Homeに戻る  一覧に戻る 

Integer Points on A^4+B^4+C^4=334562*D^4

[2026.02.15]A^4+B^4+C^4=334562*D^4の整点

■整点を求める方法は、 "A^4+B^4+C^4=3362*D^4の整点" と同様なので、詳細はそちらを参照すること。ただし、参照する数式のみ記載する。

自然数nを固定したとき、不定方程式
       A^4+B^4+C^4=2*n^2*D^4 ----------(1)
を満たす自明でない整数の組(A,B,C,D) (ただし C!=0かつgcd(A,B,C,D)=1)を探す。

以下では、Elkiesの論文(参考文献[1])の方法およびTom Womackの文書(参考文献[5])を参考にして、(1)を満たす整数の組(A,B,C,D)を探す。
ここで、整数A,B,C,Dは0以上として良い。


■x=A/C,y=B/C.t=D/Cとすると、
       x^4+y^4+1=2*n^2*t^4 ----------(2)
つまり、(2)を満たす有理数の組(x,y,t)を見つければ良い。

そのためには、nある有理数uに対して、
       ±(u^2-2)*y^2=(-u^2+4*u-2)*x^2-2*(u^2-2*u+2)*x+(-u^2+4*u-2) ----------(3a±)
       ±n*(u^2-2)*t^2=(u^2-2*u+2)*x^2+(-u^2+4*u-2)*x+(u^2-2*u+2) ----------(3b±)
の両方を満たす有理数の組(x,y,t)を見つければ良い。


■任意の有理数uについて、２次曲線(3b+)および(3b-)は、non-singularである。
また、u^2 > 2のとき、(3b+)のみ、u^2 < 2のとき、(3b-)のみが成立する。

■２次曲線(3a)がsingularであるのは、u=0,1,2のときであり。そのときに限る。
u=1のとき、(3a+)はsingularであるが、有理点を持たない。
u-0,2のとき、(3a+)はsingularであり、
       x^2 - x + 1=n*t^2　--------(**)
が有理点をもつかどうかを議論する必要がある。

334562=2*409^2であるので、以下では、n=409とする。

■n=409のとき、２次曲線(**)は、有理点(-23/15, 1/15)を持つことが確認できる。

{MAGMAでの計算]
> P2 := ProjectiveSpace(Rationals(), 2);
> N:=409;
> C := Conic(P2,-N*y^2+x^2+x*z+z^2);
> HasRationalPoint(C);
true (-23/15 : 1/15 : 1)
>


■有理数u(u!=0,1,2)の高さが小さいものから、順に調べる。
例えば、有理数uの高さが400以下の範囲で、２つの２次曲線(3a+)と(3b±)が共に有理点を持つようなuを選択すると、以下のように394個のuが抽出される。
これらのuについて、(3a+),(3b±)を共に満たす有理数の組(x,y,t)を見つければ良い。

[MAGMAによる計算]
> PP(409,1,400);
** u= 1/137 ; tau(u)= 273/136 ; -36991*x^2 + 37537*y^2 + 74530*x*z - 36991*z^2
  (40591/47221 : -3916/47221 : 1)  C1b (6564295/753673 : -306675/753673 : 1)
** u= -3/5 ; tau(u)= 13/8 ; -119*x^2 + 41*y^2 + 178*x*z - 119*z^2
  (-75/103 : -284/103 : 1)  C2b (-14920/1693 : -1173/1693 : 1)
** u= 3/61 ; tau(u)= 119/58 ; -6719*x^2 + 7433*y^2 + 14170*x*z - 6719*z^2
  (-49/279 : 314/279 : 1)  C1b (-197859/19523 : -10033/19523 : 1)
** u= 4/157 ; tau(u)= 310/153 ; -46802*x^2 + 49282*y^2 + 96116*x*z - 46802*z^2
  (-35963/301286 : 329493/301286 : 1)  C1b (-13911415/1200532 : 709673/1200532 : 1)
** u= -5/13 ; tau(u)= 31/18 ; -623*x^2 + 313*y^2 + 986*x*z - 623*z^2
  (1/5 : -6/5 : 1)  C2b (-13341/8725 : -1243/8725 : 1)
** u= -8/29 ; tau(u)= 66/37 ; -2674*x^2 + 1618*y^2 + 4420*x*z - 2674*z^2
  (19/27 : 20/27 : 1)  C2b (-4136/12985 : -183/2597 : 1)
** u= 8/53 ; tau(u)= 98/45 ; -3986*x^2 + 5554*y^2 + 9668*x*z - 3986*z^2
  (-673/3571 : -3696/3571 : 1)  C1b (-291691/79021 : -15323/79021 : 1)
** u= 8/97 ; tau(u)= 186/89 ; -15778*x^2 + 18754*y^2 + 34660*x*z - 15778*z^2
  (25953/40597 : 2456/40597 : 1)  C1b (-156232/147455 : -2463/29491 : 1)
** u= -8/109 ; tau(u)= 226/117 ; -27314*x^2 + 23698*y^2 + 51140*x*z - 27314*z^2
  (37/26 : -285/442 : 1)  C2b (110335/3992 : -94525/67864 : 1)
** u= 8/205 ; tau(u)= 402/197 ; -77554*x^2 + 83986*y^2 + 161668*x*z - 77554*z^2
  (6585/92941 : 578864/650587 : 1)  C1b (61430912/154505 : 1602693/83195 : 1)
** u= -9/89 ; tau(u)= 187/98 ; -19127*x^2 + 15761*y^2 + 35050*x*z - 19127*z^2
  (6581/9533 : 4830/9533 : 1)  C2b (-163847625/3246617 : -8636645/3246617 : 1)
** u= 9/145 ; tau(u)= 281/136 ; -36911*x^2 + 41969*y^2 + 79042*x*z - 36911*z^2
  (8495/12429 : 692/12429 : 1)  C1b (-48509/393645 : 20081/393645 : 1)
** u= 9/173 ; tau(u)= 337/164 ; -53711*x^2 + 59777*y^2 + 113650*x*z - 53711*z^2
  (1397/2143 : 9976/49289 : 1)  C1b (38800/210967 : 216935/4852241 : 1)
** u= -11/221 ; tau(u)= 453/232 ; -107527*x^2 + 97561*y^2 + 205330*x*z - 107527*z^2
  (168907/244689 : -102220/244689 : 1)  C2b (1504120/2147791 : 94785/2147791 : 1)
** u= -11/313 ; tau(u)= 637/324 ; -209831*x^2 + 195817*y^2 + 405890*x*z - 209831*z^2
  (-656881/2645161 : 3400236/2645161 : 1)  C2b (57126144/4107043 : -2774429/4107043 : 1)
** u= 12/13 ; tau(u)= 14 ; 142*x^2 + 194*y^2 + 340*x*z + 142*z^2
  (-144/119 : -67/119 : 1)  C1b (2727/316 : -133/316 : 1)
** u= 12/65 ; tau(u)= 118/53 ; -5474*x^2 + 8306*y^2 + 14068*x*z - 5474*z^2
  (214/557 : 181/557 : 1)  C1b (46916/108645 : 4559/108645 : 1)
** u= 12/149 ; tau(u)= 286/137 ; -37394*x^2 + 44258*y^2 + 81940*x*z - 37394*z^2
  (40808/70867 : 17207/70867 : 1)  C1b (26679044/13177967 : 1130377/13177967 : 1)
** u= -12/181 ; tau(u)= 374/193 ; -74354*x^2 + 65378*y^2 + 140020*x*z - 74354*z^2
  (25971/20068 : -10435/20068 : 1)  C2b (22491828/10595 : 230117/2119 : 1)
** u= 12/233 ; tau(u)= 454/221 ; -97538*x^2 + 108434*y^2 + 206260*x*z - 97538*z^2
  (510353/1740726 : 1126975/1740726 : 1)  C1b (194048356/267367735 : 2363803/53473547 : 1)
** u= 13/8 ; tau(u)= -3/5 ; 119*x^2 - 41*y^2 + 178*x*z + 119*z^2
  (-3 : -4 : 1)  C1a (-307/157 : -17/157 : 1)
** u= -13/109 ; tau(u)= 231/122 ; -29599*x^2 + 23593*y^2 + 53530*x*z - 29599*z^2
  (12953/20513 : -11642/20513 : 1)  C2b (-35112044/1222297 : 1885569/1222297 : 1)
** u= 14 ; tau(u)= 12/13 ; -142*x^2 - 194*y^2 + 340*x*z - 142*z^2
  (2/3 : -1/3 : 1)  C1a (10549/740 : -95/148 : 1)
** u= -15/113 ; tau(u)= 241/128 ; -32543*x^2 + 25313*y^2 + 58306*x*z - 32543*z^2
  (5490053/9654187 : -6040624/9654187 : 1)  C2b (-6374240/904869 : 367217/904869 : 1)
** u= 15/193 ; tau(u)= 371/178 ; -63143*x^2 + 74273*y^2 + 137866*x*z - 63143*z^2
  (-69/47 : -1858/799 : 1)  C1b (-4967060/201547 : -4099717/3426299 : 1)
** u= -16/49 ; tau(u)= 114/65 ; -8194*x^2 + 4546*y^2 + 13252*x*z - 8194*z^2
  (285/691 : 658/691 : 1)  C2b (797/62453 : -3699/62453 : 1)
** u= 19/29 ; tau(u)= 39/10 ; 161*x^2 + 1321*y^2 + 1882*x*z + 161*z^2
  (-1097/2523 : -1738/2523 : 1)  C1b (40253/64955 : -3309/64955 : 1)
** u= 20/101 ; tau(u)= 182/81 ; -12722*x^2 + 20002*y^2 + 33524*x*z - 12722*z^2
  (87385/1106348 : -788247/1106348 : 1)  C1b (-14626076/1476609 : -690193/1476609 : 1)
** u= 21/25 ; tau(u)= 29/4 ; 409*x^2 + 809*y^2 + 1282*x*z + 409*z^2
  (-581/1329 : 400/1329 : 1)  C1b (216/605 : 31/605 : 1)
** u= -21/109 ; tau(u)= 239/130 ; -33359*x^2 + 23321*y^2 + 57562*x*z - 33359*z^2
  (-34941/10387 : 52882/10387 : 1)  C2b (-321675/918644 : -62387/918644 : 1)
** u= 21/125 ; tau(u)= 229/104 ; -21191*x^2 + 30809*y^2 + 52882*x*z - 21191*z^2
  (40377/82547 : -9340/82547 : 1)  C1b (-47933800/4547427 : -2287187/4547427 : 1)
** u= -21/149 ; tau(u)= 319/170 ; -57359*x^2 + 43961*y^2 + 102202*x*z - 57359*z^2
  (-14759/26981 : 46474/26981 : 1)  C2b (2045780/548939 : -95753/548939 : 1)
** u= 21/265 ; tau(u)= 509/244 ; -118631*x^2 + 140009*y^2 + 259522*x*z - 118631*z^2
  (-3776677/8471361 : -11497624/8471361 : 1)  C1b (-456253120/327583299 : -32008019/327583299 : 1)
** u= -21/277 ; tau(u)= 575/298 ; -177167*x^2 + 153017*y^2 + 331066*x*z - 177167*z^2
  (1962561/51735293 : 53700554/51735293 : 1)  C2b (1267819033/23741815 : -64557893/23741815 : 1)
** u= 21/313 ; tau(u)= 605/292 ; -170087*x^2 + 195497*y^2 + 366466*x*z - 170087*z^2
  (-6499/26823 : 31456/26823 : 1)  C1b (11572341/501475 : -543251/501475 : 1)
** u= 23/41 ; tau(u)= 59/18 ; -119*x^2 + 2833*y^2 + 4010*x*z - 119*z^2
  (-853/73087 : 2526/10441 : 1)  C1b (-425423/14540 : -3545/2908 : 1)
** u= -23/121 ; tau(u)= 265/144 ; -40943*x^2 + 28753*y^2 + 70754*x*z - 40943*z^2
  (15773/15619 : -9768/15619 : 1)  C2b (-99847341/26954936 : -6445669/26954936 : 1)
** u= 24/53 ; tau(u)= 82/29 ; -1106*x^2 + 5042*y^2 + 7300*x*z - 1106*z^2
  (-309/763 : 100/109 : 1)  C1b (154525/15317 : 6435/15317 : 1)
** u= 24/125 ; tau(u)= 226/101 ; -19826*x^2 + 30674*y^2 + 51652*x*z - 19826*z^2
  (-696/1915 : -15539/13405 : 1)  C1b (-2101/67784 : -589/12824 : 1)
** u= 24/145 ; tau(u)= 266/121 ; -28706*x^2 + 41474*y^2 + 71332*x*z - 28706*z^2
  (-3581/1007771 : 842116/1007771 : 1)  C1b (-32237/35520 : 2603/35520 : 1)
** u= -25/49 ; tau(u)= 123/74 ; -10327*x^2 + 4177*y^2 + 15754*x*z - 10327*z^2
  (4143/1483 : -4970/1483 : 1)  C2b (-2461/1445 : 243/1445 : 1)
** u= 27/61 ; tau(u)= 95/34 ; -1583*x^2 + 6713*y^2 + 9754*x*z - 1583*z^2
  (-55701/69427 : -605702/485989 : 1)  C1b (104900/6909 : 30683/48363 : 1)
** u= -27/181 ; tau(u)= 389/208 ; -85799*x^2 + 64793*y^2 + 152050*x*z - 85799*z^2
  (19551/32089 : -19936/32089 : 1)  C2b (16792904/3990047 : 798593/3990047 : 1)
** u= -27/185 ; tau(u)= 397/212 ; -89159*x^2 + 67721*y^2 + 158338*x*z - 89159*z^2
  (-8725/936857 : 1083864/936857 : 1)  C2b (10177623/22441673 : -1000187/22441673 : 1)
** u= -27/245 ; tau(u)= 517/272 ; -147239*x^2 + 119321*y^2 + 268018*x*z - 147239*z^2
  (-2075/10483 : 13776/10483 : 1)  C2b (-79097984/41649175 : -5646469/41649175 : 1)
** u= 27/277 ; tau(u)= 527/250 ; -124271*x^2 + 152729*y^2 + 278458*x*z - 124271*z^2
  (98447/3095723 : 2692590/3095723 : 1)  C1b (5752557/4057700 : -253019/4057700 : 1)
** u= 28/289 ; tau(u)= 550/261 ; -135458*x^2 + 166258*y^2 + 303284*x*z - 135458*z^2
  (-9791/235604 : 222513/235604 : 1)  C1b (3526844/5634231 : -242789/5634231 : 1)
** u= 28/389 ; tau(u)= 750/361 ; -259858*x^2 + 301858*y^2 + 563284*x*z - 259858*z^2
  (139877/410078 : 233605/410078 : 1)  C1b (22177319/8231396 : 944631/8231396 : 1)
** u= 29/4 ; tau(u)= 21/25 ; -409*x^2 - 809*y^2 + 1282*x*z - 409*z^2
  (89/237 : 32/237 : 1)  C1a (1663/1047 : 73/1047 : 1)
** u= 31/18 ; tau(u)= -5/13 ; 623*x^2 - 313*y^2 + 986*x*z + 623*z^2
  (-3899/5287 : 4578/5287 : 1)  C1a (6540/10807 : -979/10807 : 1)
** u= -31/73 ; tau(u)= 177/104 ; -20671*x^2 + 9697*y^2 + 32290*x*z - 20671*z^2
  (20317/15223 : 18548/15223 : 1)  C2b (-30966319/618647 : 2001231/618647 : 1)
** u= -32/233 ; tau(u)= 498/265 ; -139426*x^2 + 107554*y^2 + 249028*x*z - 139426*z^2
  (-11707/101 : -13432/101 : 1)  C2b (77615/2044367 : 8193/157259 : 1)
** u= 33/41 ; tau(u)= 49/8 ; 961*x^2 + 2273*y^2 + 3490*x*z + 961*z^2
  (-8707/29007 : 28/29007 : 1)  C1b (60008/39865 : -699/7973 : 1)
** u= -33/65 ; tau(u)= 163/98 ; -18119*x^2 + 7361*y^2 + 27658*x*z - 18119*z^2
  (11/3 : -14/3 : 1)  C2b (-132594431/8081404 : 9354469/8081404 : 1)
** u= -33/145 ; tau(u)= 323/178 ; -62279*x^2 + 40961*y^2 + 105418*x*z - 62279*z^2
  (-2007/93241 : -117074/93241 : 1)  C2b (296305/320999 : -15711/320999 : 1)
** u= 33/149 ; tau(u)= 265/116 ; -25823*x^2 + 43313*y^2 + 71314*x*z - 25823*z^2
  (60287/239769 : 112436/239769 : 1)  C1b (9464/18175 : 771/18175 : 1)
** u= 33/265 ; tau(u)= 497/232 ; -106559*x^2 + 139361*y^2 + 248098*x*z - 106559*z^2
  (84745/576899 : -415852/576899 : 1)  C1b (-270616/2927669 : -142459/2927669 : 1)
** u= -36/185 ; tau(u)= 406/221 ; -96386*x^2 + 67154*y^2 + 166132*x*z - 96386*z^2
  (76/129 : -89/129 : 1)  C2b (-9205399/1020585 : -540739/1020585 : 1)
** u= 36/221 ; tau(u)= 406/185 ; -67154*x^2 + 96386*y^2 + 166132*x*z - 67154*z^2
  (9769/365832 : 295213/365832 : 1)  C1b (-16792475/7934029 : 976613/7934029 : 1)
** u= -36/221 ; tau(u)= 478/257 ; -130802*x^2 + 96386*y^2 + 229780*x*z - 130802*z^2
  (495086/534371 : 299073/534371 : 1)  C2b (3964236/1368313 : -182369/1368313 : 1)
** u= 36/277 ; tau(u)= 518/241 ; -114866*x^2 + 152162*y^2 + 269620*x*z - 114866*z^2
  (392/1707 : 1063/1707 : 1)  C1b (-825529/2743308 : -148183/2743308 : 1)
** u= 36/349 ; tau(u)= 662/313 ; -194642*x^2 + 242306*y^2 + 439540*x*z - 194642*z^2
  (-111886/3288863 : 3060465/3288863 : 1)  C1b (228748052/62246841 : -9831767/62246841 : 1)
** u= 39/10 ; tau(u)= 19/29 ; -161*x^2 - 1321*y^2 + 1882*x*z - 161*z^2
  (3113/23499 : -854/3357 : 1)  C1a (6731/14020 : -627/14020 : 1)
** u= 39/49 ; tau(u)= 59/10 ; 1321*x^2 + 3281*y^2 + 5002*x*z + 1321*z^2
  (-911/573 : 574/573 : 1)  C1b (1102325/746764 : 64143/746764 : 1)
** u= 39/89 ; tau(u)= 139/50 ; -3479*x^2 + 14321*y^2 + 20842*x*z - 3479*z^2
  (127177/768221 : 71486/768221 : 1)  C1b (-918179/207579 : -41107/207579 : 1)
** u= -40/113 ; tau(u)= 266/153 ; -45218*x^2 + 23938*y^2 + 72356*x*z - 45218*z^2
  (22085/63709 : -65844/63709 : 1)  C2b (-550777/33288 : -34781/33288 : 1)
** u= -41/49 ; tau(u)= 139/90 ; -14519*x^2 + 3121*y^2 + 21002*x*z - 14519*z^2
  (155/547 : -966/547 : 1)  C2b (-332675/63204 : 34387/63204 : 1)
** u= 41/113 ; tau(u)= 185/72 ; -8687*x^2 + 23857*y^2 + 35906*x*z - 8687*z^2
  (36263/795373 : -432924/795373 : 1)  C1b (-340200/15781 : 14771/15781 : 1)
** u= 41/221 ; tau(u)= 401/180 ; -63119*x^2 + 96001*y^2 + 162482*x*z - 63119*z^2
  (-149261/1311977 : 1215648/1311977 : 1)  C1b (-247754305/229917112 : -18109379/229917112 : 1)
** u= -43/101 ; tau(u)= 245/144 ; -39623*x^2 + 18553*y^2 + 61874*x*z - 39623*z^2
  (15263/22201 : -20496/22201 : 1)  C2b (169184/34251 : 9551/34251 : 1)
** u= 44/61 ; tau(u)= 78/17 ; 1358*x^2 + 5506*y^2 + 8020*x*z + 1358*z^2
  (-538/3053 : 149/3053 : 1)  C1b (921004/8245 : -7785/1649 : 1)
** u= 44/89 ; tau(u)= 134/45 ; -2114*x^2 + 13906*y^2 + 19892*x*z - 2114*z^2
  (202/4781 : -207/683 : 1)  C1b (-115/196 : 4063/80164 : 1)
** u= 44/125 ; tau(u)= 206/81 ; -11186*x^2 + 29314*y^2 + 44372*x*z - 11186*z^2
  (446/11911 : -6795/11911 : 1)  C1b (1823053/225780 : 76553/225780 : 1)
** u= 45/49 ; tau(u)= 53/4 ; 1993*x^2 + 2777*y^2 + 4834*x*z + 1993*z^2
  (-209/309 : -112/309 : 1)  C1b (137064/8155 : 6487/8155 : 1)
** u= -45/61 ; tau(u)= 167/106 ; -20447*x^2 + 5417*y^2 + 29914*x*z - 20447*z^2
  (-8795/21013 : -54582/21013 : 1)  C2b (-33509053/254652 : 2767579/254652 : 1)
** u= 45/373 ; tau(u)= 701/328 ; -213143*x^2 + 276233*y^2 + 493426*x*z - 213143*z^2
  (14213/1055447 : 912636/1055447 : 1)  C1b (9222392/4577655 : 389671/4577655 : 1)
** u= 48/101 ; tau(u)= 154/53 ; -3314*x^2 + 18098*y^2 + 26020*x*z - 3314*z^2
  (849/6619 : 274/6619 : 1)  C1b (10812928/934609 : 450011/934609 : 1)
** u= 48/109 ; tau(u)= 170/61 ; -5138*x^2 + 21458*y^2 + 31204*x*z - 5138*z^2
  (279/3353 : 166/479 : 1)  C1b (-227147/97512 : -11027/97512 : 1)
** u= 48/169 ; tau(u)= 290/121 ; -26978*x^2 + 54818*y^2 + 86404*x*z - 26978*z^2
  (566/13173 : -8591/13173 : 1)  C1b (1081824/414325 : -45239/414325 : 1)
** u= -48/377 ; tau(u)= 802/425 ; -358946*x^2 + 281954*y^2 + 645508*x*z - 358946*z^2
  (-7994/1469 : -10535/1469 : 1)  C2b (2557293680/212626351 : 129373203/212626351 : 1)
** u= 49/8 ; tau(u)= 33/41 ; -961*x^2 - 2273*y^2 + 3490*x*z - 961*z^2
  (3021/9041 : -1876/9041 : 1)  C1a (264329/124505 : 2247/24901 : 1)
** u= 49/89 ; tau(u)= 129/40 ; -799*x^2 + 13441*y^2 + 19042*x*z - 799*z^2
  (-1383/98507 : 27748/98507 : 1)  C1b (-5361056/555865 : -225267/555865 : 1)
** u= 49/337 ; tau(u)= 625/288 ; -163487*x^2 + 224737*y^2 + 393026*x*z - 163487*z^2
  (175237/934163 : -609000/934163 : 1)  C1b (-119519112/64613567 : 7293913/64613567 : 1)
** u= 51/245 ; tau(u)= 439/194 ; -72671*x^2 + 117449*y^2 + 195322*x*z - 72671*z^2
  (-51589/237483 : -238574/237483 : 1)  C1b (-96546884/36419675 : 5192949/36419675 : 1)
** u= -51/373 ; tau(u)= 797/424 ; -356951*x^2 + 275657*y^2 + 637810*x*z - 356951*z^2
  (-140711/5982891 : -6951604/5982891 : 1)  C2b (58576840720/21981989 : -3114721665/21981989 : 1)
** u= 53/4 ; tau(u)= 45/49 ; -1993*x^2 - 2777*y^2 + 4834*x*z - 1993*z^2
  (209/393 : -28/393 : 1)  C1a (-291691/79021 : -15323/79021 : 1)
** u= 57/109 ; tau(u)= 161/52 ; -2159*x^2 + 20513*y^2 + 29170*x*z - 2159*z^2
  (-21933/71317 : 53012/71317 : 1)  C1b (1298357/1188139 : -70623/1188139 : 1)
** u= 57/205 ; tau(u)= 353/148 ; -40559*x^2 + 80801*y^2 + 127858*x*z - 40559*z^2
  (-122887/1426975 : -8003252/9988825 : 1)  C1b (-932728/702607 : 410691/4918249 : 1)
** u= 57/377 ; tau(u)= 697/320 ; -201551*x^2 + 281009*y^2 + 489058*x*z - 201551*z^2
  (667/1269 : 32/1269 : 1)  C1b (176232755/40852589 : 7550087/40852589 : 1)
** u= 59/10 ; tau(u)= 39/49 ; -1321*x^2 - 3281*y^2 + 5002*x*z - 1321*z^2
  (1025/347 : 266/347 : 1)  C1a (-44119/27865 : 2507/27865 : 1)
** u= 59/18 ; tau(u)= 23/41 ; 119*x^2 - 2833*y^2 + 4010*x*z + 119*z^2
  (1/4111 : 846/4111 : 1)  C1a (-149707/69101 : 6781/69101 : 1)
** u= -59/121 ; tau(u)= 301/180 ; -61319*x^2 + 25801*y^2 + 94082*x*z - 61319*z^2
  (-9041/15797 : 36168/15797 : 1)  C2b (-190593/7417088 : 503599/7417088 : 1)
** u= -59/233 ; tau(u)= 525/292 ; -167047*x^2 + 105097*y^2 + 279106*x*z - 167047*z^2
  (-71397/3150937 : -4048000/3150937 : 1)  C2b (48010159/24413185 : -2205291/24413185 : 1)
** u= -60/53 ; tau(u)= 166/113 ; -21938*x^2 + 2018*y^2 + 31156*x*z - 21938*z^2
  (-358/885 : 3847/885 : 1)  C2b (1249465/450481 : 135407/450481 : 1)
** u= -60/229 ; tau(u)= 518/289 ; -163442*x^2 + 101282*y^2 + 271924*x*z - 163442*z^2
  (2/15 : -17/15 : 1)  C2b (591804/583555 : -30077/583555 : 1)
** u= 61/113 ; tau(u)= 165/52 ; -1687*x^2 + 21817*y^2 + 30946*x*z - 1687*z^2
  (-38355/358771 : -24572/51253 : 1)  C1b (27515/410432 : 17067/410432 : 1)
** u= -61/173 ; tau(u)= 407/234 ; -105791*x^2 + 56137*y^2 + 169370*x*z - 105791*z^2
  (101747/110299 : -1890/2251 : 1)  C2b (10784510765/738378924 : -627931585/738378924 : 1)
** u= 63/137 ; tau(u)= 211/74 ; -6983*x^2 + 33569*y^2 + 48490*x*z - 6983*z^2
  (-13579/49089 : 38762/49089 : 1)  C1b (495377844/37505593 : -20648953/37505593 : 1)
** u= -63/149 ; tau(u)= 361/212 ; -85919*x^2 + 40433*y^2 + 134290*x*z - 85919*z^2
  (21599/23849 : 22116/23849 : 1)  C2b (6421267/1840221 : -346081/1840221 : 1)
** u= -63/325 ; tau(u)= 713/388 ; -297119*x^2 + 207281*y^2 + 512338*x*z - 297119*z^2
  (110491/12241 : -119880/12241 : 1)  C2b (266160/24701749 : 1349411/24701749 : 1)
** u= 66/37 ; tau(u)= -8/29 ; 2674*x^2 - 1618*y^2 + 4420*x*z + 2674*z^2
  (-172/151 : 125/151 : 1)  C1a (-2632/18961 : -1011/18961 : 1)
** u= -67/145 ; tau(u)= 357/212 ; -85399*x^2 + 37561*y^2 + 131938*x*z - 85399*z^2
  (35/81 : -88/81 : 1)  C2b (6241159/9093760 : -454839/9093760 : 1)
** u= -68/125 ; tau(u)= 318/193 ; -69874*x^2 + 26626*y^2 + 105748*x*z - 69874*z^2
  (-64/71 : 205/71 : 1)  C2b (338180/728513 : 39387/728513 : 1)
** u= 69/109 ; tau(u)= 149/40 ; 1561*x^2 + 19001*y^2 + 26962*x*z + 1561*z^2
  (-7999/71755 : 19652/71755 : 1)  C1b (129288/215725 : -10739/215725 : 1)
** u= 69/137 ; tau(u)= 205/68 ; -4487*x^2 + 32777*y^2 + 46786*x*z - 4487*z^2
  (-1541/36689 : 16288/36689 : 1)  C1b (-635752/203647 : 28653/203647 : 1)
** u= 72/169 ; tau(u)= 266/97 ; -13634*x^2 + 51938*y^2 + 75940*x*z - 13634*z^2
  (-69/1972 : -65/116 : 1)  C1b (1601768/499101 : 67217/499101 : 1)
** u= -72/281 ; tau(u)= 634/353 ; -244034*x^2 + 152738*y^2 + 407140*x*z - 244034*z^2
  (7168/25017 : -24577/25017 : 1)  C2b (-9437627/1911141 : -610417/1911141 : 1)
** u= -75/181 ; tau(u)= 437/256 ; -125447*x^2 + 59897*y^2 + 196594*x*z - 125447*z^2
  (19307/37694037 : 54528800/37694037 : 1)  C2b (-4742363/11913792 : -974209/11913792 : 1)
** u= -75/317 ; tau(u)= 709/392 ; -301703*x^2 + 195353*y^2 + 508306*x*z - 301703*z^2
  (25/3 : 28/3 : 1)  C2b (41024869/3016339 : 2217043/3016339 : 1)
** u= 76/233 ; tau(u)= 390/157 ; -43522*x^2 + 102802*y^2 + 157876*x*z - 43522*z^2
  (25637/324764 : 1255013/2273348 : 1)  C1b (-403859/104140 : 134667/728980 : 1)
** u= -77/85 ; tau(u)= 247/162 ; -46559*x^2 + 8521*y^2 + 66938*x*z - 46559*z^2
  (7/11 : -18/11 : 1)  C2b (-3366380/1551999 : 449179/1551999 : 1)
** u= -77/373 ; tau(u)= 823/450 ; -399071*x^2 + 272329*y^2 + 683258*x*z - 399071*z^2
  (790087/1179083 : -784050/1179083 : 1)  C2b (-301443260/240586971 : 26759753/240586971 : 1)
** u= 78/17 ; tau(u)= 44/61 ; -1358*x^2 - 5506*y^2 + 8020*x*z - 1358*z^2
  (84/481 : -7/481 : 1)  C1a (7568420/520471 : -316395/520471 : 1)
** u= 79/97 ; tau(u)= 115/18 ; 5593*x^2 + 12577*y^2 + 19466*x*z + 5593*z^2
  (-14045/39899 : -498/2347 : 1)  C1b (37389148/2166245 : 1655903/2166245 : 1)
** u= -79/121 ; tau(u)= 321/200 ; -73759*x^2 + 23041*y^2 + 109282*x*z - 73759*z^2
  (9157/1303 : -14740/1303 : 1)  C2b (10340752/4008265 : -623499/4008265 : 1)
** u= -79/281 ; tau(u)= 641/360 ; -252959*x^2 + 151681*y^2 + 417122*x*z - 252959*z^2
  (11845/53759 : -57468/53759 : 1)  C2b (-299520/3709547 : 226039/3709547 : 1)
** u= -80/73 ; tau(u)= 226/153 ; -40418*x^2 + 4258*y^2 + 57476*x*z - 40418*z^2
  (-10601/64901 : -224358/64901 : 1)  C2b (-1090149/155131 : -154703/155131 : 1)
** u= 81/277 ; tau(u)= 473/196 ; -70271*x^2 + 146897*y^2 + 230290*x*z - 70271*z^2
  (-8423/52327 : -45108/52327 : 1)  C1b (-2205418952/323816235 : 20354701/64763247 : 1)
** u= 82/29 ; tau(u)= 24/53 ; 1106*x^2 - 5042*y^2 + 7300*x*z + 1106*z^2
  (-953/7296 : 1345/7296 : 1)  C1a (345973/179296 : 17379/179296 : 1)
** u= 83/389 ; tau(u)= 695/306 ; -180383*x^2 + 295753*y^2 + 489914*x*z - 180383*z^2
  (566321/1742905 : 642894/1742905 : 1)  C1b (4959467/3430468 : 219703/3430468 : 1)
** u= -84/61 ; tau(u)= 206/145 ; -34994*x^2 + 386*y^2 + 49492*x*z - 34994*z^2
  (5/22 : -179/22 : 1)  C2b (173763/657172 : -217061/657172 : 1)
** u= -84/65 ; tau(u)= 214/149 ; -37346*x^2 + 1394*y^2 + 52852*x*z - 37346*z^2
  (-4968/6581 : -55327/6581 : 1)  C2b (285132/236687 : -44093/236687 : 1)
** u= -84/85 ; tau(u)= 254/169 ; -50066*x^2 + 7394*y^2 + 71572*x*z - 50066*z^2
  (-28148/174983 : -510263/174983 : 1)  C2b (8428751/2765495 : 738439/2765495 : 1)
** u= 84/101 ; tau(u)= 118/17 ; 6478*x^2 + 13346*y^2 + 20980*x*z + 6478*z^2
  (-1641/4744 : 83/4744 : 1)  C1b (-369169/419127 : 20393/419127 : 1)
** u= -84/197 ; tau(u)= 478/281 ; -150866*x^2 + 70562*y^2 + 235540*x*z - 150866*z^2
  (9561/17276 : 16799/17276 : 1)  C2b (-104005/435756 : -32495/435756 : 1)
** u= 84/269 ; tau(u)= 454/185 ; -61394*x^2 + 137666*y^2 + 213172*x*z - 61394*z^2
  (-15723/256364 : 188843/256364 : 1)  C1b (177641/299505 : -13097/299505 : 1)
** u= -84/337 ; tau(u)= 758/421 ; -347426*x^2 + 220082*y^2 + 581620*x*z - 347426*z^2
  (9749/358 : -11875/358 : 1)  C2b (12776948/5116259 : 599731/5116259 : 1)
** u= -88/373 ; tau(u)= 834/461 ; -417298*x^2 + 270514*y^2 + 703300*x*z - 417298*z^2
  (1705186/158618359 : 195225667/158618359 : 1)  C2b (10787456/2398553 : 539031/2398553 : 1)
** u= 89/205 ; tau(u)= 321/116 ; -18991*x^2 + 76129*y^2 + 110962*x*z - 18991*z^2
  (-8063/2055 : -6436/2055 : 1)  C1b (-25382795/860611 : -1078269/860611 : 1)
** u= 92/157 ; tau(u)= 222/65 ; 14*x^2 + 40834*y^2 + 57748*x*z + 14*z^2
  (-3276/113 : -721/113 : 1)  C1b (-444460/539879 : 29073/539879 : 1)
** u= -93/269 ; tau(u)= 631/362 ; -253439*x^2 + 136073*y^2 + 406810*x*z - 253439*z^2
  (-51731/1155909 : 11443058/8091363 : 1)  C2b (-37780/209851 : 99555/1468957 : 1)
** u= 95/34 ; tau(u)= 27/61 ; 1583*x^2 - 6713*y^2 + 9754*x*z + 1583*z^2
  (-7/67 : 138/469 : 1)  C1a (10573/9693 : 4561/67851 : 1)
** u= -95/229 ; tau(u)= 553/324 ; -200927*x^2 + 95857*y^2 + 314834*x*z - 200927*z^2
  (-535861/4925845 : -7754436/4925845 : 1)  C2b (287137/106920 : 14819/106920 : 1)
** u= 96/137 ; tau(u)= 178/41 ; 5854*x^2 + 28322*y^2 + 40900*x*z + 5854*z^2
  (-197/267 : 27448/31773 : 1)  C1b (34491016/8193497 : 10757073/57354479 : 1)
** u= -96/349 ; tau(u)= 794/445 ; -386834*x^2 + 234386*y^2 + 639652*x*z - 386834*z^2
  (4055/14929 : 15158/14929 : 1)  C2b (2307275/285051 : 9539/21927 : 1)
** u= 96/365 ; tau(u)= 634/269 ; -135506*x^2 + 257234*y^2 + 411172*x*z - 135506*z^2
  (3450/11503 : -3541/11503 : 1)  C1b (2474720/380231 : -105061/380231 : 1)
** u= -96/365 ; tau(u)= 826/461 ; -415826*x^2 + 257234*y^2 + 691492*x*z - 415826*z^2
  (3697/13136 : 13057/13136 : 1)  C2b (-49317623/33633712 : -4273933/33633712 : 1)
** u= 97/293 ; tau(u)= 489/196 ; -67423*x^2 + 162289*y^2 + 248530*x*z - 67423*z^2
  (-6700891/261172731 : 176173508/261172731 : 1)  C1b (1823117/2305160 : 21819/461032 : 1)
** u= 98/45 ; tau(u)= 8/53 ; 3986*x^2 - 5554*y^2 + 9668*x*z + 3986*z^2
  (-2764/11243 : -6489/11243 : 1)  C1a (137064/8155 : 6487/8155 : 1)
** u= -99/113 ; tau(u)= 325/212 ; -80087*x^2 + 15737*y^2 + 115426*x*z - 80087*z^2
  (-2111/271 : -5220/271 : 1)  C2b (688775/518217 : -47123/518217 : 1)
** u= 105/157 ; tau(u)= 209/52 ; 5617*x^2 + 38273*y^2 + 54706*x*z + 5617*z^2
  (-36651/207235 : -66004/207235 : 1)  C1b (-139208/105405 : -6929/105405 : 1)
** u= -105/193 ; tau(u)= 491/298 ; -166583*x^2 + 63473*y^2 + 252106*x*z - 166583*z^2
  (222867/77315 : -278578/77315 : 1)  C2b (13073821/3618380 : -768087/3618380 : 1)
** u= 105/377 ; tau(u)= 649/272 ; -136943*x^2 + 273233*y^2 + 432226*x*z - 136943*z^2
  (68977/490479 : 263512/490479 : 1)  C1b (59591720/56639749 : -2989197/56639749 : 1)
** u= -108/97 ; tau(u)= 302/205 ; -72386*x^2 + 7154*y^2 + 102868*x*z - 72386*z^2
  (205/1162 : -22857/8134 : 1)  C2b (-3591459/126340 : -3416699/884380 : 1)
** u= 108/149 ; tau(u)= 190/41 ; 8302*x^2 + 32738*y^2 + 47764*x*z + 8302*z^2
  (-5315/1654 : 2229/1654 : 1)  C1b (-428193/96580 : 17833/96580 : 1)
** u= -108/149 ; tau(u)= 406/257 ; -120434*x^2 + 32738*y^2 + 176500*x*z - 120434*z^2
  (-53483/8344 : 114825/8344 : 1)  C2b (8072412/831109 : 611303/831109 : 1)
** u= 108/257 ; tau(u)= 406/149 ; -32738*x^2 + 120434*y^2 + 176500*x*z - 32738*z^2
  (-634/9211 : -5633/9211 : 1)  C1b (417057/433916 : 22993/433916 : 1)
** u= -108/289 ; tau(u)= 686/397 ; -303554*x^2 + 155378*y^2 + 482260*x*z - 303554*z^2
  (257/2 : -357/2 : 1)  C2b (132060804/22012507 : -7357087/22012507 : 1)
** u= 108/373 ; tau(u)= 638/265 ; -128786*x^2 + 266594*y^2 + 418708*x*z - 128786*z^2
  (-24164/8563 : -25347/8563 : 1)  C1b (13595875/693019 : 587117/693019 : 1)
** u= -109/349 ; tau(u)= 807/458 ; -407647*x^2 + 231721*y^2 + 663130*x*z - 407647*z^2
  (-59297/109783 : 1501622/768481 : 1)  C2b (-2819188/1325849 : 1565757/9280943 : 1)
** u= 111/337 ; tau(u)= 563/226 ; -89831*x^2 + 214817*y^2 + 329290*x*z - 89831*z^2
  (-2454701/1329891 : 2874614/1329891 : 1)  C1b (443181204/5289343 : 19123709/5289343 : 1)
** u= -112/145 ; tau(u)= 402/257 ; -119554*x^2 + 29506*y^2 + 174148*x*z - 119554*z^2
  (17337/532423 : -1046582/532423 : 1)  C2b (2602448/2226995 : 165033/2226995 : 1)
** u= 112/229 ; tau(u)= 346/117 ; -14834*x^2 + 92338*y^2 + 132260*x*z - 14834*z^2
  (4639/174896 : 61281/174896 : 1)  C1b (4154336/1113153 : -174853/1113153 : 1)
** u= -112/257 ; tau(u)= 626/369 ; -259778*x^2 + 119554*y^2 + 404420*x*z - 259778*z^2
  (-7373/25301 : 1064010/581923 : 1)  C2b (392591/158695 : 93203/729997 : 1)
** u= 114/65 ; tau(u)= -16/49 ; 8194*x^2 - 4546*y^2 + 13252*x*z + 8194*z^2
  (29/153 : -14/9 : 1)  C1a (-62453/797 : -3699/797 : 1)
** u= 115/18 ; tau(u)= 79/97 ; -5593*x^2 - 12577*y^2 + 19466*x*z - 5593*z^2
  (3061/9475 : -894/9475 : 1)  C1a (36458053/10063140 : -1519103/10063140 : 1)
** u= 117/389 ; tau(u)= 661/272 ; -134279*x^2 + 288953*y^2 + 450610*x*z - 134279*z^2
  (12817/39327 : -21016/275289 : 1)  C1b (1680573/489035 : -98065/684649 : 1)
** u= 118/17 ; tau(u)= 84/101 ; -6478*x^2 - 13346*y^2 + 20980*x*z - 6478*z^2
  (503/558 : -409/558 : 1)  C1a (21098556/575423 : 917443/575423 : 1)
** u= 118/53 ; tau(u)= 12/65 ; 5474*x^2 - 8306*y^2 + 14068*x*z + 5474*z^2
  (-12336/57065 : -32467/57065 : 1)  C1a (-371911500/3562501 : -16860811/3562501 : 1)
** u= 119/58 ; tau(u)= 3/61 ; 6719*x^2 - 7433*y^2 + 14170*x*z + 6719*z^2
  (3389/10719 : -13546/10719 : 1)  C1a (-2410177/156055 : -22595/31211 : 1)
** u= 119/337 ; tau(u)= 555/218 ; -80887*x^2 + 212977*y^2 + 322186*x*z - 80887*z^2
  (-22785/385867 : 264662/385867 : 1)  C1b (-37388/14425 : 1863/14425 : 1)
** u= -120/197 ; tau(u)= 514/317 ; -186578*x^2 + 63218*y^2 + 278596*x*z - 186578*z^2
  (-11493/38591 : -11728/5513 : 1)  C2b (7175039/1363855 : 465573/1363855 : 1)
** u= 120/241 ; tau(u)= 362/121 ; -14882*x^2 + 101762*y^2 + 145444*x*z - 14882*z^2
  (-2786/1653 : 2849/1653 : 1)  C1b (12554525/935752 : -522231/935752 : 1)
** u= -120/241 ; tau(u)= 602/361 ; -246242*x^2 + 101762*y^2 + 376804*x*z - 246242*z^2
  (1213991/1548267 : -1551464/1548267 : 1)  C2b (1120872/345877 : -62729/345877 : 1)
** u= -120/269 ; tau(u)= 658/389 ; -288242*x^2 + 130322*y^2 + 447364*x*z - 288242*z^2
  (-81/206 : 407/206 : 1)  C2b (-1594040/1374441 : -174467/1374441 : 1)
** u= 121/125 ; tau(u)= 129/4 ; 14609*x^2 + 16609*y^2 + 31282*x*z + 14609*z^2
  (-3535/2603 : 616/2603 : 1)  C1b (-346288/140897 : -14727/140897 : 1)
** u= -121/289 ; tau(u)= 699/410 ; -321559*x^2 + 152401*y^2 + 503242*x*z - 321559*z^2
  (4179/149563 : 212534/149563 : 1)  C2b (-4967132/562945 : -339561/562945 : 1)
** u= 123/74 ; tau(u)= -25/49 ; 10327*x^2 - 4177*y^2 + 15754*x*z + 10327*z^2
  (7153/104007 : -172270/104007 : 1)  C1a (3206804/279625 : -230643/279625 : 1)
** u= -123/389 ; tau(u)= 901/512 ; -509159*x^2 + 287513*y^2 + 826930*x*z - 509159*z^2
  (-536087/598263 : -1437088/598263 : 1)  C2b (-9603103/3629991 : -722161/3629991 : 1)
** u= -124/317 ; tau(u)= 758/441 ; -373586*x^2 + 185602*y^2 + 589940*x*z - 373586*z^2
  (-45359/10058 : 76125/10058 : 1)  C2b (-1290305988/1578547465 : 32357783/315709493 : 1)
** u= 129/4 ; tau(u)= 121/125 ; -14609*x^2 - 16609*y^2 + 31282*x*z - 14609*z^2
  (19631/26239 : -5060/26239 : 1)  C1a (836120/751909 : -39489/751909 : 1)
** u= 129/40 ; tau(u)= 49/89 ; 799*x^2 - 13441*y^2 + 19042*x*z + 799*z^2
  (-349/27159 : -5516/27159 : 1)  C1a (5478595/5881067 : -341421/5881067 : 1)
** u= 132/221 ; tau(u)= 310/89 ; 1582*x^2 + 80258*y^2 + 113524*x*z + 1582*z^2
  (-2962/210739 : -2713/210739 : 1)  C1b (513845/927253 : 44343/927253 : 1)
** u= 132/365 ; tau(u)= 598/233 ; -91154*x^2 + 249026*y^2 + 375028*x*z - 91154*z^2
  (-52879/1008 : -4747/144 : 1)  C1b (-15135/772108 : -33293/772108 : 1)
** u= -133/157 ; tau(u)= 447/290 ; -150511*x^2 + 31609*y^2 + 217498*x*z - 150511*z^2
  (-506251/59374565 : -130362878/59374565 : 1)  C2b (10734095/686108 : 942039/686108 : 1)
** u= 133/365 ; tau(u)= 597/232 ; -89959*x^2 + 248761*y^2 + 374098*x*z - 89959*z^2
  (2701/18679 : -7276/18679 : 1)  C1b (2843638795/1791079789 : 127516821/1791079789 : 1)
** u= 134/45 ; tau(u)= 44/89 ; 2114*x^2 - 13906*y^2 + 19892*x*z + 2114*z^2
  (-275/32566 : -12183/32566 : 1)  C1a (-699/356 : -12827/145604 : 1)
** u= 135/337 ; tau(u)= 539/202 ; -63383*x^2 + 208913*y^2 + 308746*x*z - 63383*z^2
  (6479/369547 : -194698/369547 : 1)  C1b (1466398043/74301045 : 61758479/74301045 : 1)
** u= 139/50 ; tau(u)= 39/89 ; 3479*x^2 - 14321*y^2 + 20842*x*z + 3479*z^2
  (-1268581/7616849 : -649970/7616849 : 1)  C1a (-142452/34325 : -5941/34325 : 1)
** u= 139/90 ; tau(u)= -41/49 ; 14519*x^2 - 3121*y^2 + 21002*x*z + 14519*z^2
  (2479/26137 : -60354/26137 : 1)  C1a (-311287/6993 : 27799/6993 : 1)
** u= 139/197 ; tau(u)= 255/58 ; 12593*x^2 + 58297*y^2 + 84346*x*z + 12593*z^2
  (-14107/72105 : -17482/72105 : 1)  C1b (312218620/12798061 : -13221879/12798061 : 1)
** u= -140/197 ; tau(u)= 534/337 ; -207538*x^2 + 58018*y^2 + 304756*x*z - 207538*z^2
  (692/269 : 997/269 : 1)  C2b (-7145335/859556 : 621543/859556 : 1)
** u= -141/109 ; tau(u)= 359/250 ; -105119*x^2 + 3881*y^2 + 148762*x*z - 105119*z^2
  (659131/7844039 : 5496070/1120577 : 1)  C2b (10551020/3138883 : 1866819/3138883 : 1)
** u= 143/313 ; tau(u)= 483/170 ; -37351*x^2 + 175489*y^2 + 253738*x*z - 37351*z^2
  (-31/557 : 302/557 : 1)  C1b (796863671/5694785 : 33467937/5694785 : 1)
** u= -144/145 ; tau(u)= 434/289 ; -146306*x^2 + 21314*y^2 + 209092*x*z - 146306*z^2
  (8/9 : 17/9 : 1)  C2b (-811240/3115701 : 408331/3115701 : 1)
** u= -144/149 ; tau(u)= 442/293 ; -150962*x^2 + 23666*y^2 + 216100*x*z - 150962*z^2
  (-1389/1154 : -5953/1154 : 1)  C2b (1163944/817275 : -17599/163455 : 1)
** u= 144/185 ; tau(u)= 226/41 ; 17374*x^2 + 47714*y^2 + 71812*x*z + 17374*z^2
  (-2419/1580 : 1647/1580 : 1)  C1b (269323640/45072557 : -12165683/45072557 : 1)
** u= -144/313 ; tau(u)= 770/457 ; -396962*x^2 + 175202*y^2 + 613636*x*z - 396962*z^2
  (1130309/971195 : 1089486/971195 : 1)  C2b (-1928807/8011632 : 613483/8011632 : 1)
** u= 147/277 ; tau(u)= 407/130 ; -12191*x^2 + 131849*y^2 + 187258*x*z - 12191*z^2
  (-21949/2049967 : -672686/2049967 : 1)  C1b (1735141/716699 : -76407/716699 : 1)
** u= -147/305 ; tau(u)= 757/452 ; -386999*x^2 + 164441*y^2 + 594658*x*z - 386999*z^2
  (-11597/9 : 302624/153 : 1)  C2b (384471/548641 : -471863/9326897 : 1)
** u= 147/317 ; tau(u)= 487/170 ; -36191*x^2 + 179369*y^2 + 258778*x*z - 36191*z^2
  (-135621/1476535 : 855862/1476535 : 1)  C1b (-29861092/6121335 : 1314889/6121335 : 1)
** u= -148/221 ; tau(u)= 590/369 ; -250418*x^2 + 75778*y^2 + 370004*x*z - 250418*z^2
  (7/1634 : 2961/1634 : 1)  C2b (2773125/620917 : -185183/620917 : 1)
** u= 149/40 ; tau(u)= 69/109 ; -1561*x^2 - 19001*y^2 + 26962*x*z - 1561*z^2
  (1103/18579 : -788/18579 : 1)  C1a (1846553/453048 : -78121/453048 : 1)
** u= -151/173 ; tau(u)= 497/324 ; -187151*x^2 + 37057*y^2 + 269810*x*z - 187151*z^2
  (35063/5610133 : 12550968/5610133 : 1)  C2b (3800872/14955285 : 238115/2991057 : 1)
** u= 153/269 ; tau(u)= 385/116 ; -3503*x^2 + 121313*y^2 + 171634*x*z - 3503*z^2
  (8037/480391 : 34688/480391 : 1)  C1b (-707667432/32187017 : 29488403/32187017 : 1)
** u= 154/53 ; tau(u)= 48/101 ; 3314*x^2 - 18098*y^2 + 26020*x*z + 3314*z^2
  (-32/1731 : -685/1731 : 1)  C1a (-2312496/889043 : 99337/889043 : 1)
** u= 156/397 ; tau(u)= 638/241 ; -91826*x^2 + 290882*y^2 + 431380*x*z - 91826*z^2
  (-45681/615134 : -402221/615134 : 1)  C1b (-53334284/2702773 : 2298167/2702773 : 1)
** u= -159/113 ; tau(u)= 385/272 ; -122687*x^2 + 257*y^2 + 173506*x*z - 122687*z^2
  (179/109 : 2792/109 : 1)  C2b (-445849/356129 : 673841/356129 : 1)
** u= -159/181 ; tau(u)= 521/340 ; -205919*x^2 + 40241*y^2 + 296722*x*z - 205919*z^2
  (8013/153559 : 47792/21937 : 1)  C2b (36432800/733849 : 3411063/733849 : 1)
** u= 159/289 ; tau(u)= 419/130 ; -8519*x^2 + 141761*y^2 + 200842*x*z - 8519*z^2
  (-33217/1682491 : 499358/1682491 : 1)  C1b (-138964892/4017055 : 5792821/4017055 : 1)
** u= 161/52 ; tau(u)= 57/109 ; 2159*x^2 - 20513*y^2 + 29170*x*z + 2159*z^2
  (-717/10423 : -928/10423 : 1)  C1a (-1359528/158719 : 56581/158719 : 1)
** u= 161/169 ; tau(u)= 177/8 ; 25793*x^2 + 31201*y^2 + 57250*x*z + 25793*z^2
  (-47169/42661 : -18668/42661 : 1)  C1b (-27535699/216488 : 1299549/216488 : 1)
** u= 163/98 ; tau(u)= -33/65 ; 18119*x^2 - 7361*y^2 + 27658*x*z + 18119*z^2
  (-3/11 : 14/11 : 1)  C1a (-495995/123108 : -28847/123108 : 1)
** u= 165/52 ; tau(u)= 61/113 ; 1687*x^2 - 21817*y^2 + 30946*x*z + 1687*z^2
  (-79315/2559901 : 468196/2559901 : 1)  C1a (71771411/5016451 : -3007311/5016451 : 1)
** u= -165/173 ; tau(u)= 511/338 ; -201263*x^2 + 32633*y^2 + 288346*x*z - 201263*z^2
  (15/11 : -26/11 : 1)  C2b (716115932/806815 : 74369289/806815 : 1)
** u= -165/221 ; tau(u)= 607/386 ; -270767*x^2 + 70457*y^2 + 395674*x*z - 270767*z^2
  (-32133/13463 : -84226/13463 : 1)  C2b (112680775/31274708 : -7795581/31274708 : 1)
** u= -165/389 ; tau(u)= 943/554 ; -586607*x^2 + 275417*y^2 + 916474*x*z - 586607*z^2
  (1253/3709 : 70462/63053 : 1)  C2b (-4827373/1009132 : 5993763/17155244 : 1)
** u= 166/113 ; tau(u)= -60/53 ; 21938*x^2 - 2018*y^2 + 31156*x*z + 21938*z^2
  (138/1229 : -4387/1229 : 1)  C1a (19288668/3364255 : 2993429/3364255 : 1)
** u= 167/106 ; tau(u)= -45/61 ; 20447*x^2 - 5417*y^2 + 29914*x*z + 20447*z^2
  (-2485/7921 : 12306/7921 : 1)  C1a (27292220/252033 : 2256641/252033 : 1)
** u= 168/169 ; tau(u)= 170 ; 28222*x^2 + 28898*y^2 + 57124*x*z + 28222*z^2
  (-1331/1230 : 169/1230 : 1)  C1b (-239175/284521 : 13091/284521 : 1)
** u= -168/229 ; tau(u)= 626/397 ; -286994*x^2 + 76658*y^2 + 420100*x*z - 286994*z^2
  (7999569/7947233 : -11297500/7947233 : 1)  C2b (-200052536/5803547 : -16702503/5803547 : 1)
** u= 168/241 ; tau(u)= 314/73 ; 17566*x^2 + 87938*y^2 + 126820*x*z + 17566*z^2
  (-427819/533889 : -485696/533889 : 1)  C1b (68025872/131709689 : -6566447/131709689 : 1)
** u= 170 ; tau(u)= 168/169 ; -28222*x^2 - 28898*y^2 + 57124*x*z - 28222*z^2
  (6991/8154 : 143/8154 : 1)  C1a (-6624/350251 : 17381/350251 : 1)
** u= 170/61 ; tau(u)= 48/109 ; 5138*x^2 - 21458*y^2 + 31204*x*z + 5138*z^2
  (1830/24961 : -14711/24961 : 1)  C1a (-983909/185960 : 40931/185960 : 1)
** u= 172/289 ; tau(u)= 406/117 ; 2206*x^2 + 137458*y^2 + 194420*x*z + 2206*z^2
  (-4744/204169 : 26469/204169 : 1)  C1b (313215/501596 : 24715/501596 : 1)
** u= 175/377 ; tau(u)= 579/202 ; -50983*x^2 + 253633*y^2 + 365866*x*z - 50983*z^2
  (6353/122289 : -43514/122289 : 1)  C1b (-7468577908/42191987 : -313883109/42191987 : 1)
** u= 177/8 ; tau(u)= 161/169 ; -25793*x^2 - 31201*y^2 + 57250*x*z - 25793*z^2
  (5209/7661 : -1508/7661 : 1)  C1a (-189737048/1215881 : 9012351/1215881 : 1)
** u= 177/104 ; tau(u)= -31/73 ; 20671*x^2 - 9697*y^2 + 32290*x*z + 20671*z^2
  (135463/95547 : -318868/95547 : 1)  C1a (-734072/456551 : -35991/456551 : 1)
** u= 178/41 ; tau(u)= 96/137 ; -5854*x^2 - 28322*y^2 + 40900*x*z - 5854*z^2
  (257/47 : -6872/5593 : 1)  C1a (4499/392 : 22299/46648 : 1)
** u= -180/181 ; tau(u)= 542/361 ; -228242*x^2 + 33122*y^2 + 326164*x*z - 228242*z^2
  (10/9 : 19/9 : 1)  C2b (136643/12020 : 14101/12020 : 1)
** u= 180/349 ; tau(u)= 518/169 ; -24722*x^2 + 211202*y^2 + 300724*x*z - 24722*z^2
  (13996/224177 : -37921/224177 : 1)  C1b (26912411/13439195 : 1213097/13439195 : 1)
** u= 180/377 ; tau(u)= 574/197 ; -45218*x^2 + 251858*y^2 + 361876*x*z - 45218*z^2
  (-377554/19777245 : -8998811/19777245 : 1)  C1b (1174729428/787097423 : -55734029/787097423 : 1)
** u= -180/377 ; tau(u)= 934/557 ; -588098*x^2 + 251858*y^2 + 904756*x*z - 588098*z^2
  (85297/41356 : 13023/5908 : 1)  C2b (397140/2623807 : 158041/2623807 : 1)
** u= 182/81 ; tau(u)= 20/101 ; 12722*x^2 - 20002*y^2 + 33524*x*z + 12722*z^2
  (-1223368/15579101 : -11107647/15579101 : 1)  C1a (28545060/1132171 : 1312403/1132171 : 1)
** u= 183/265 ; tau(u)= 347/82 ; 20041*x^2 + 106961*y^2 + 153898*x*z + 20041*z^2
  (-138157/70387 : -97402/70387 : 1)  C1b (4463204/89355 : 187711/89355 : 1)
** u= -183/361 ; tau(u)= 905/544 ; -558383*x^2 + 227153*y^2 + 852514*x*z - 558383*z^2
  (-196515/83659 : 416936/83659 : 1)  C2b (-71718333/6353765 : 5150027/6353765 : 1)
** u= 184/229 ; tau(u)= 274/45 ; 29806*x^2 + 71026*y^2 + 108932*x*z + 29806*z^2
  (-15053/38144 : 13227/38144 : 1)  C1b (450872/590865 : 36181/590865 : 1)
** u= 184/305 ; tau(u)= 426/121 ; 4574*x^2 + 152194*y^2 + 215332*x*z + 4574*z^2
  (-2138/53013 : 60929/371091 : 1)  C1b (95/848 : -249/5936 : 1)
** u= 185/72 ; tau(u)= 41/113 ; 8687*x^2 - 23857*y^2 + 35906*x*z + 8687*z^2
  (1369/11917 : -516/701 : 1)  C1a (269323640/45072557 : -12165683/45072557 : 1)
** u= 186/89 ; tau(u)= 8/97 ; 15778*x^2 - 18754*y^2 + 34660*x*z + 15778*z^2
  (-7389/18817 : -9320/18817 : 1)  C1a (-1570120/265657 : 69795/265657 : 1)
** u= 187/98 ; tau(u)= -9/89 ; 19127*x^2 - 15761*y^2 + 35050*x*z + 19127*z^2
  (10243/32237 : -46074/32237 : 1)  C1a (-17933/5417324 : 281953/5417324 : 1)
** u= -188/245 ; tau(u)= 678/433 ; -339634*x^2 + 84706*y^2 + 495028*x*z - 339634*z^2
  (87336/1217185 : -2312933/1217185 : 1)  C2b (-12131675/5336989 : 1374669/5336989 : 1)
** u= -188/353 ; tau(u)= 894/541 ; -550018*x^2 + 213874*y^2 + 834580*x*z - 550018*z^2
  (13153/38548 : 47831/38548 : 1)  C2b (2462555/4683716 : 247095/4683716 : 1)
** u= -189/221 ; tau(u)= 631/410 ; -300479*x^2 + 61961*y^2 + 433882*x*z - 300479*z^2
  (-5523/55061 : 130306/55061 : 1)  C2b (1137643165/7275867 : -104793139/7275867 : 1)
** u= 189/257 ; tau(u)= 325/68 ; 26473*x^2 + 96377*y^2 + 141346*x*z + 26473*z^2
  (-3975/20411 : -464/20411 : 1)  C1b (4357189/1077891 : 198181/1077891 : 1)
** u= 189/317 ; tau(u)= 445/128 ; 2953*x^2 + 165257*y^2 + 233746*x*z + 2953*z^2
  (-1913/128521 : 7248/128521 : 1)  C1b (-1077303/2679328 : -119557/2679328 : 1)
** u= -189/373 ; tau(u)= 935/562 ; -595967*x^2 + 242537*y^2 + 909946*x*z - 595967*z^2
  (162683/29651 : -221574/29651 : 1)  C2b (3531508/6115857 : -128685673/2501385513 : 1)
** u= 190/41 ; tau(u)= 108/149 ; -8302*x^2 - 32738*y^2 + 47764*x*z - 8302*z^2
  (2030/2341 : -2121/2341 : 1)  C1a (-5118793455/5354428 : 216219277/5354428 : 1)
** u= -192/257 ; tau(u)= 706/449 ; -366338*x^2 + 95234*y^2 + 535300*x*z - 366338*z^2
  (2163/591568 : 1157149/591568 : 1)  C2b (24934864/16623983 : -1507557/16623983 : 1)
** u= -195/197 ; tau(u)= 589/392 ; -269303*x^2 + 39593*y^2 + 384946*x*z - 269303*z^2
  (2867/6849 : 230636/116433 : 1)  C2b (765320/331509 : 1081691/5635653 : 1)
** u= -196/157 ; tau(u)= 510/353 ; -210802*x^2 + 10882*y^2 + 298516*x*z - 210802*z^2
  (-2010/11 : -8881/11 : 1)  C2b (223340/411617 : 54777/411617 : 1)
** u= 196/221 ; tau(u)= 246/25 ; 37166*x^2 + 59266*y^2 + 98932*x*z + 37166*z^2
  (-39301/67526 : 24535/67526 : 1)  C1b (800084812/6484213 : -36254121/6484213 : 1)
** u= 200/373 ; tau(u)= 546/173 ; -19858*x^2 + 238258*y^2 + 338116*x*z - 19858*z^2
  (-388049/33036456 : 10448465/33036456 : 1)  C1b (-476339384/4355281 : -19851813/4355281 : 1)
** u= 201/205 ; tau(u)= 209/4 ; 40369*x^2 + 43649*y^2 + 84082*x*z + 40369*z^2
  (-384575/380943 : -105884/380943 : 1)  C1b (-108360/305149 : 13121/305149 : 1)
** u= -201/377 ; tau(u)= 955/578 ; -627767*x^2 + 243857*y^2 + 952426*x*z - 627767*z^2
  (175097/7177 : -272306/7177 : 1)  C2b (-20689476/17061025 : 2378101/17061025 : 1)
** u= 203/325 ; tau(u)= 447/122 ; 11441*x^2 + 170041*y^2 + 241018*x*z + 11441*z^2
  (-13970637/57478735 : 30046246/57478735 : 1)  C1b (3346900/9932491 : 442521/9932491 : 1)
** u= 203/365 ; tau(u)= 527/162 ; -11279*x^2 + 225241*y^2 + 318938*x*z - 11279*z^2
  (7759/491471 : 81846/491471 : 1)  C1b (-22072252/71321745 : 3137113/71321745 : 1)
** u= 204/241 ; tau(u)= 278/37 ; 38878*x^2 + 74546*y^2 + 118900*x*z + 38878*z^2
  (-9318/7051 : 5795/7051 : 1)  C1b (249804/17737 : -11311/17737 : 1)
** u= -204/241 ; tau(u)= 686/445 ; -354434*x^2 + 74546*y^2 + 512212*x*z - 354434*z^2
  (-4559/11832 : -33691/11832 : 1)  C2b (1291655/2189476 : 146307/2189476 : 1)
** u= -204/245 ; tau(u)= 694/449 ; -361586*x^2 + 78434*y^2 + 523252*x*z - 361586*z^2
  (39285/1492 : -82061/1492 : 1)  C2b (-3633435/250852 : 344173/250852 : 1)
** u= -204/277 ; tau(u)= 758/481 ; -421106*x^2 + 111842*y^2 + 616180*x*z - 421106*z^2
  (98389/159444 : 213869/159444 : 1)  C2b (-189983095/44057977 : 18254475/44057977 : 1)
** u= -204/337 ; tau(u)= 878/541 ; -543746*x^2 + 185522*y^2 + 812500*x*z - 543746*z^2
  (-37282/162901 : 329315/162901 : 1)  C2b (-140136412/5835169 : 10544391/5835169 : 1)
** u= 205/68 ; tau(u)= 69/137 ; 4487*x^2 - 32777*y^2 + 46786*x*z + 4487*z^2
  (6413/1337 : -608/191 : 1)  C1a (6616409/4676360 : -353421/4676360 : 1)
** u= 206/81 ; tau(u)= 44/125 ; 11186*x^2 - 29314*y^2 + 44372*x*z + 11186*z^2
  (263/24032 : -15165/24032 : 1)  C1a (-110921313/529505 : 4765813/529505 : 1)
** u= 206/145 ; tau(u)= -84/61 ; 34994*x^2 - 386*y^2 + 49492*x*z + 34994*z^2
  (-134/165 : 1123/165 : 1)  C1a (11089/8279 : 7097/8279 : 1)
** u= -207/229 ; tau(u)= 665/436 ; -337343*x^2 + 62033*y^2 + 485074*x*z - 337343*z^2
  (15593/14639 : 26508/14639 : 1)  C2b (11893168/3568525 : -953723/3568525 : 1)
** u= 207/233 ; tau(u)= 259/26 ; 41497*x^2 + 65729*y^2 + 109930*x*z + 41497*z^2
  (-20479/38877 : -10610/38877 : 1)  C1b (861852/1179835 : 15407/235967 : 1)
** u= 207/305 ; tau(u)= 403/98 ; 23641*x^2 + 143201*y^2 + 205258*x*z + 23641*z^2
  (-704495/4258213 : -1106574/4258213 : 1)  C1b (-14943613/847740 : 622453/847740 : 1)
** u= -207/313 ; tau(u)= 833/520 ; -497951*x^2 + 153089*y^2 + 736738*x*z - 497951*z^2
  (-12821/22873 : 60396/22873 : 1)  C2b (719211/4768384 : -330967/4768384 : 1)
** u= -208/233 ; tau(u)= 674/441 ; -345698*x^2 + 65314*y^2 + 497540*x*z - 345698*z^2
  (3566/2377 : 97125/40409 : 1)  C2b (4782261/1057717 : -6756031/17981189 : 1)
** u= 209/4 ; tau(u)= 201/205 ; -40369*x^2 - 43649*y^2 + 84082*x*z - 40369*z^2
  (19569/24337 : -3932/24337 : 1)  C1a (-6030065/417424 : 302871/417424 : 1)
** u= 209/52 ; tau(u)= 105/157 ; -5617*x^2 - 38273*y^2 + 54706*x*z - 5617*z^2
  (8511/2711 : -4612/2711 : 1)  C1a (-1168854655/1921313337 : -98201519/1921313337 : 1)
** u= -209/241 ; tau(u)= 691/450 ; -361319*x^2 + 72481*y^2 + 521162*x*z - 361319*z^2
  (-201731/2023661 : -4853130/2023661 : 1)  C2b (52094820/5765629 : 4526213/5765629 : 1)
** u= 209/245 ; tau(u)= 281/36 ; 41089*x^2 + 76369*y^2 + 122642*x*z + 41089*z^2
  (-152125/231853 : -123564/231853 : 1)  C1b (-26567651552/2739990455 : -1142233417/2739990455 : 1)
** u= 211/74 ; tau(u)= 63/137 ; 6983*x^2 - 33569*y^2 + 48490*x*z + 6983*z^2
  (19/1641 : 778/1641 : 1)  C1a (120949/32717 : 5453/32717 : 1)
** u= 214/149 ; tau(u)= -84/65 ; 37346*x^2 - 1394*y^2 + 52852*x*z + 37346*z^2
  (15904/14089 : 143491/14089 : 1)  C1a (-504835/95483 : -95277/95483 : 1)
** u= -216/277 ; tau(u)= 770/493 ; -439442*x^2 + 106802*y^2 + 639556*x*z - 439442*z^2
  (-6209/1411 : -14808/1411 : 1)  C2b (213773/121677 : -13463/121677 : 1)
** u= 216/361 ; tau(u)= 506/145 ; 4606*x^2 + 213986*y^2 + 302692*x*z + 4606*z^2
  (-991/60873 : 2356/60873 : 1)  C1b (883592/451635 : 41519/451635 : 1)
** u= 217/281 ; tau(u)= 345/64 ; 38897*x^2 + 110833*y^2 + 166114*x*z + 38897*z^2
  (-73081/59817 : 58496/59817 : 1)  C1b (80460584/9627011 : 3563181/9627011 : 1)
** u= -220/289 ; tau(u)= 798/509 ; -469762*x^2 + 118642*y^2 + 685204*x*z - 469762*z^2
  (43410/55021 : -75191/55021 : 1)  C2b (45694516/9803875 : -3329223/9803875 : 1)
** u= 222/65 ; tau(u)= 92/157 ; -14*x^2 - 40834*y^2 + 57748*x*z - 14*z^2
  (113/1992 : -563/1992 : 1)  C1a (14750668/2494475 : -622011/2494475 : 1)
** u= -223/353 ; tau(u)= 929/576 ; -613823*x^2 + 199489*y^2 + 912770*x*z - 613823*z^2
  (568391/890857 : 1057920/890857 : 1)  C2b (4331760/1683503 : -256805/1683503 : 1)
** u= 226/41 ; tau(u)= 144/185 ; -17374*x^2 - 47714*y^2 + 71812*x*z - 17374*z^2
  (28831/8180 : 5283/8180 : 1)  C1a (-340200/15781 : 14771/15781 : 1)
** u= 226/101 ; tau(u)= 24/125 ; 19826*x^2 - 30674*y^2 + 51652*x*z + 19826*z^2
  (2491/11829 : 84020/82803 : 1)  C1a (53192/53895 : 28291/377265 : 1)
** u= 226/117 ; tau(u)= -8/109 ; 27314*x^2 - 23698*y^2 + 51140*x*z + 27314*z^2
  (179/1061 : -22452/18037 : 1)  C1a (2432347/134496 : 2188819/2286432 : 1)
** u= 226/153 ; tau(u)= -80/73 ; 40418*x^2 - 4258*y^2 + 57476*x*z + 40418*z^2
  (-11639/149917 : 437118/149917 : 1)  C1a (-91680/320923 : -33991/320923 : 1)
** u= 229/104 ; tau(u)= 21/125 ; 21191*x^2 - 30809*y^2 + 52882*x*z + 21191*z^2
  (-8519/45909 : 28780/45909 : 1)  C1a (-859335/247553 : -36337/247553 : 1)
** u= 231/122 ; tau(u)= -13/109 ; 29599*x^2 - 23593*y^2 + 53530*x*z + 29599*z^2
  (22367/19249 : -45490/19249 : 1)  C1a (111591245/5286508 : 6035475/5286508 : 1)
** u= 231/265 ; tau(u)= 299/34 ; 51049*x^2 + 87089*y^2 + 142762*x*z + 51049*z^2
  (-84177/198079 : 13438/198079 : 1)  C1b (63961860/6669161 : 2983129/6669161 : 1)
** u= -231/361 ; tau(u)= 953/592 ; -647567*x^2 + 207281*y^2 + 961570*x*z - 647567*z^2
  (-12917/11213 : 39824/11213 : 1)  C2b (199199776/32377397 : -13480113/32377397 : 1)
** u= -231/365 ; tau(u)= 961/596 ; -657071*x^2 + 213089*y^2 + 976882*x*z - 657071*z^2
  (127579/2718345 : -4609328/2718345 : 1)  C2b (14045/30896 : 729779/12636464 : 1)
** u= 233/313 ; tau(u)= 393/80 ; 41489*x^2 + 141649*y^2 + 208738*x*z + 41489*z^2
  (-139813/550805 : 137464/550805 : 1)  C1b (-8620328/5714923 : 396771/5714923 : 1)
** u= 235/317 ; tau(u)= 399/82 ; 41777*x^2 + 145753*y^2 + 214426*x*z + 41777*z^2
  (-3069/13199 : -2638/13199 : 1)  C1b (1433935669/139262395 : 62231967/139262395 : 1)
** u= 236/373 ; tau(u)= 510/137 ; 18158*x^2 + 222562*y^2 + 315796*x*z + 18158*z^2
  (-28465/430998 : 46757/430998 : 1)  C1b (856189844/43558031 : 35809311/43558031 : 1)
** u= 239/130 ; tau(u)= -21/109 ; 33359*x^2 - 23321*y^2 + 57562*x*z + 33359*z^2
  (-3865/29873 : -31826/29873 : 1)  C1a (1252260/419141 : 84247/419141 : 1)
** u= -240/289 ; tau(u)= 818/529 ; -502082*x^2 + 109442*y^2 + 726724*x*z - 502082*z^2
  (1344773/1304551 : -2110618/1304551 : 1)  C2b (-3149615/709689 : -331193/709689 : 1)
** u= -240/293 ; tau(u)= 826/533 ; -510578*x^2 + 114098*y^2 + 739876*x*z - 510578*z^2
  (-149/35948 : 76273/35948 : 1)  C2b (-29532328/5865605 : 3013621/5865605 : 1)
** u= 241/128 ; tau(u)= -15/113 ; 32543*x^2 - 25313*y^2 + 58306*x*z + 32543*z^2
  (-3359/54225 : 58096/54225 : 1)  C1a (-641640/270593 : -28631/270593 : 1)
** u= 245/144 ; tau(u)= -43/101 ; 39623*x^2 - 18553*y^2 + 61874*x*z + 39623*z^2
  (-22201/15263 : 20496/15263 : 1)  C1a (-41408480/2513571 : 2544199/2513571 : 1)
** u= 246/25 ; tau(u)= 196/221 ; -37166*x^2 - 59266*y^2 + 98932*x*z - 37166*z^2
  (586/611 : 385/611 : 1)  C1a (-954628/1165225 : 79599/1165225 : 1)
** u= 247/162 ; tau(u)= -77/85 ; 46559*x^2 - 8521*y^2 + 66938*x*z + 46559*z^2
  (1279/27073 : -65466/27073 : 1)  C1a (662932/476205 : -103033/476205 : 1)
** u= 248/289 ; tau(u)= 330/41 ; 58142*x^2 + 105538*y^2 + 170404*x*z + 58142*z^2
  (-2053/3710 : -221/530 : 1)  C1b (443017/3430600 : 160203/3430600 : 1)
** u= 248/397 ; tau(u)= 546/149 ; 17102*x^2 + 253714*y^2 + 359620*x*z + 17102*z^2
  (-399/6569 : -892/6569 : 1)  C1b (84967637/2677664 : 3543981/2677664 : 1)
** u= -249/241 ; tau(u)= 731/490 ; -418199*x^2 + 54161*y^2 + 596362*x*z - 418199*z^2
  (12151/57034727 : -158460694/57034727 : 1)  C2b (-1574092/556225 : 232479/556225 : 1)
** u= -252/241 ; tau(u)= 734/493 ; -422594*x^2 + 52658*y^2 + 602260*x*z - 422594*z^2
  (-58327/251648 : 838683/251648 : 1)  C2b (-28411380/6354041 : -3923395/6354041 : 1)
** u= 252/317 ; tau(u)= 382/65 ; 55054*x^2 + 137474*y^2 + 209428*x*z + 55054*z^2
  (-1630/2001 : 1517/2001 : 1)  C1b (6267567/4487780 : -371761/4487780 : 1)
** u= 252/353 ; tau(u)= 454/101 ; 43102*x^2 + 185714*y^2 + 269620*x*z + 43102*z^2
  (-231848/292351 : 257085/292351 : 1)  C1b (6692029/3069588 : 19319/180564 : 1)
** u= -253/205 ; tau(u)= 663/458 ; -355519*x^2 + 20041*y^2 + 503578*x*z - 355519*z^2
  (12711/4975 : -289994/34825 : 1)  C2b (-9796/1565 : 5498103/4480595 : 1)
** u= 254/169 ; tau(u)= -84/85 ; 50066*x^2 - 7394*y^2 + 71572*x*z + 50066*z^2
  (-6/7 : -13/7 : 1)  C1a (86644/204065 : -29569/204065 : 1)
** u= 255/58 ; tau(u)= 139/197 ; -12593*x^2 - 58297*y^2 + 84346*x*z - 12593*z^2
  (19549/8943 : -12374/8943 : 1)  C1a (77188/1461923 : 61083/1461923 : 1)
** u= 256/337 ; tau(u)= 418/81 ; 52414*x^2 + 161602*y^2 + 240260*x*z + 52414*z^2
  (-739/2044 : -81/196 : 1)  C1b (38880/53371 : 21655/373597 : 1)
** u= 259/26 ; tau(u)= 207/233 ; -41497*x^2 - 65729*y^2 + 109930*x*z - 41497*z^2
  (91547/49069 : 26490/49069 : 1)  C1a (-269812876/42272883 : -13042379/42272883 : 1)
** u= 265/116 ; tau(u)= 33/149 ; 25823*x^2 - 43313*y^2 + 71314*x*z + 25823*z^2
  (-11163/509 : -8068/509 : 1)  C1a (28010903/5274608 : -1360383/5274608 : 1)
** u= 265/144 ; tau(u)= -23/121 ; 40943*x^2 - 28753*y^2 + 70754*x*z + 40943*z^2
  (-103721/225271 : 173448/225271 : 1)  C1a (-773595/13154117 : -697777/13154117 : 1)
** u= 265/337 ; tau(u)= 409/72 ; 59857*x^2 + 156913*y^2 + 237506*x*z + 59857*z^2
  (-104717/268001 : 104388/268001 : 1)  C1b (-1640240720/691108587 : 69316591/691108587 : 1)
** u= 266/97 ; tau(u)= 72/169 ; 13634*x^2 - 51938*y^2 + 75940*x*z + 13634*z^2
  (-42803/239623 : 23608/239623 : 1)  C1a (5134845/758051 : 225065/758051 : 1)
** u= 266/121 ; tau(u)= 24/145 ; 28706*x^2 - 41474*y^2 + 71332*x*z + 28706*z^2
  (-19883/40419 : -4708/40419 : 1)  C1a (-485048/130943 : 20573/130943 : 1)
** u= 266/153 ; tau(u)= -40/113 ; 45218*x^2 - 23938*y^2 + 72356*x*z + 45218*z^2
  (-7265/3817 : 6588/3817 : 1)  C1a (-6979944/1264739 : 381007/1264739 : 1)
** u= 268/349 ; tau(u)= 430/81 ; 58702*x^2 + 171778*y^2 + 256724*x*z + 58702*z^2
  (-27067/96988 : 21411/96988 : 1)  C1b (3393973/996957 : 12329/76689 : 1)
** u= 272/281 ; tau(u)= 290/9 ; 73822*x^2 + 83938*y^2 + 158084*x*z + 73822*z^2
  (-17749/13153 : 462/1879 : 1)  C1b (-13605880/7023849 : 578191/7023849 : 1)
** u= -272/293 ; tau(u)= 858/565 ; -564466*x^2 + 97714*y^2 + 810148*x*z - 564466*z^2
  (2235/88124 : -207983/88124 : 1)  C2b (-113313767/1266640 : 11497221/1266640 : 1)
** u= 272/337 ; tau(u)= 402/65 ; 65534*x^2 + 153154*y^2 + 235588*x*z + 65534*z^2
  (-80289/56333 : -53302/56333 : 1)  C1b (-326137/702647 : 29811/702647 : 1)
** u= 273/136 ; tau(u)= 1/137 ; 36991*x^2 - 37537*y^2 + 74530*x*z + 36991*z^2
  (-1241/1059 : -116/1059 : 1)  C1a (-636832/1012369 : -43803/1012369 : 1)
** u= 273/281 ; tau(u)= 289/8 ; 74401*x^2 + 83393*y^2 + 158050*x*z + 74401*z^2
  (-114127/159483 : -13532/159483 : 1)  C1b (-72055299/40776352 : -3077761/40776352 : 1)
** u= 274/45 ; tau(u)= 184/229 ; -29806*x^2 - 71026*y^2 + 108932*x*z - 29806*z^2
  (4066/11839 : -2841/11839 : 1)  C1a (-45679112/2375653 : -2008511/2375653 : 1)
** u= 276/317 ; tau(u)= 358/41 ; 72814*x^2 + 124802*y^2 + 204340*x*z + 72814*z^2
  (-133116/312571 : 4003/44653 : 1)  C1b (-9024819/806893 : -392353/806893 : 1)
** u= 278/37 ; tau(u)= 204/241 ; -38878*x^2 - 74546*y^2 + 118900*x*z - 38878*z^2
  (190769/211358 : 21205/30194 : 1)  C1a (103937100/16255621 : -4407385/16255621 : 1)
** u= -280/377 ; tau(u)= 1034/657 ; -784898*x^2 + 205858*y^2 + 1147556*x*z - 784898*z^2
  (-503797/7300231 : 14988912/7300231 : 1)  C2b (40546960/1630571 : 3256637/1630571 : 1)
** u= 280/389 ; tau(u)= 498/109 ; 54638*x^2 + 224242*y^2 + 326404*x*z + 54638*z^2
  (-74661/322027 : -91492/322027 : 1)  C1b (-43462471/1849625 : 1821963/1849625 : 1)
** u= 281/36 ; tau(u)= 209/245 ; -41089*x^2 - 76369*y^2 + 122642*x*z - 41089*z^2
  (10589/27533 : 84/27533 : 1)  C1a (4581205/2553453 : 196507/2553453 : 1)
** u= 281/136 ; tau(u)= 9/145 ; 36911*x^2 - 41969*y^2 + 79042*x*z + 36911*z^2
  (44229/18209 : -59404/18209 : 1)  C1a (-13605880/7023849 : 578191/7023849 : 1)
** u= -285/229 ; tau(u)= 743/514 ; -447167*x^2 + 23657*y^2 + 633274*x*z - 447167*z^2
  (-17091/134785 : -91538/19255 : 1)  C2b (-10822193/1650292 : 2178821/1650292 : 1)
** u= 286/137 ; tau(u)= 12/149 ; 37394*x^2 - 44258*y^2 + 81940*x*z + 37394*z^2
  (-37642/58173 : 1513/58173 : 1)  C1a (-1103897/450321 : -46799/450321 : 1)
** u= 287/313 ; tau(u)= 339/26 ; 81017*x^2 + 113569*y^2 + 197290*x*z + 81017*z^2
  (-81/43 : -2410/14491 : 1)  C1b (-2404/11315 : 32793/762631 : 1)
** u= -288/277 ; tau(u)= 842/565 ; -555506*x^2 + 70514*y^2 + 791908*x*z - 555506*z^2
  (5436/2537 : -11341/2537 : 1)  C2b (43145965/2707317 : 4838251/2707317 : 1)
** u= -288/353 ; tau(u)= 994/641 ; -738818*x^2 + 166274*y^2 + 1070980*x*z - 738818*z^2
  (-29547/4373 : -69256/4373 : 1)  C2b (11853512/2836071 : 896927/2836071 : 1)
** u= 289/8 ; tau(u)= 273/281 ; -74401*x^2 - 83393*y^2 + 158050*x*z - 74401*z^2
  (22857/26597 : 7412/26597 : 1)  C1a (6737496/635587 : 311077/635587 : 1)
** u= 290/9 ; tau(u)= 272/281 ; -73822*x^2 - 83938*y^2 + 158084*x*z - 73822*z^2
  (5905/8581 : 66/8581 : 1)  C1a (-48509/393645 : 20081/393645 : 1)
** u= 290/121 ; tau(u)= 48/169 ; 26978*x^2 - 54818*y^2 + 86404*x*z + 26978*z^2
  (-6971/20166 : 1573/20166 : 1)  C1a (134400/163829 : 10631/163829 : 1)
** u= -297/241 ; tau(u)= 779/538 ; -490679*x^2 + 27953*y^2 + 695050*x*z - 490679*z^2
  (-587/2157 : -10918/2157 : 1)  C2b (-15596796/1429057 : 2900537/1429057 : 1)
** u= 297/313 ; tau(u)= 329/16 ; 87697*x^2 + 107729*y^2 + 196450*x*z + 87697*z^2
  (-14483/9427 : 2424/9427 : 1)  C1b (878704/3175837 : -172511/3175837 : 1)
** u= 299/34 ; tau(u)= 231/265 ; -51049*x^2 - 87089*y^2 + 142762*x*z - 51049*z^2
  (15571/11653 : 8702/11653 : 1)  C1a (-530412/5842687 : 270953/5842687 : 1)
** u= 301/180 ; tau(u)= -59/121 ; 61319*x^2 - 25801*y^2 + 94082*x*z + 61319*z^2
  (-85567/357905 : -458436/357905 : 1)  C1a (6926224/2502969 : 585247/2502969 : 1)
** u= 302/205 ; tau(u)= -108/97 ; 72386*x^2 - 7154*y^2 + 102868*x*z + 72386*z^2
  (-244/399 : 6313/2793 : 1)  C1a (-9015/18188 : 12497/127316 : 1)
** u= -308/277 ; tau(u)= 862/585 ; -589586*x^2 + 58594*y^2 + 837908*x*z - 589586*z^2
  (25595/19595536 : 62101353/19595536 : 1)  C2b (9550172/12536835 : 1181761/12536835 : 1)
** u= -309/269 ; tau(u)= 847/578 ; -572687*x^2 + 49241*y^2 + 812890*x*z - 572687*z^2
  (3053/7113 : 18394/7113 : 1)  C2b (-10155460/1125309 : -1559455/1125309 : 1)
** u= 309/361 ; tau(u)= 413/52 ; 90073*x^2 + 165161*y^2 + 266050*x*z + 90073*z^2
  (-79219/127767 : -63080/127767 : 1)  C1b (9895384/503577 : -447181/503577 : 1)
** u= 310/89 ; tau(u)= 132/221 ; -1582*x^2 - 80258*y^2 + 113524*x*z - 1582*z^2
  (254/18195 : 101/18195 : 1)  C1a (8620175/9339332 : -524823/9339332 : 1)
** u= 310/153 ; tau(u)= 4/157 ; 46802*x^2 - 49282*y^2 + 96116*x*z + 46802*z^2
  (-347/1870 : -1473/1870 : 1)  C1a (-1742401/2140169 : -97471/2140169 : 1)
** u= 314/73 ; tau(u)= 168/241 ; -17566*x^2 - 87938*y^2 + 126820*x*z - 17566*z^2
  (46273/306027 : -35872/306027 : 1)  C1a (-9860091/3940640 : 93329/788128 : 1)
** u= 318/193 ; tau(u)= -68/125 ; 69874*x^2 - 26626*y^2 + 105748*x*z + 69874*z^2
  (-931/4786 : -6685/4786 : 1)  C1a (-3827108/80195 : 263013/80195 : 1)
** u= 319/170 ; tau(u)= -21/149 ; 57359*x^2 - 43961*y^2 + 102202*x*z + 57359*z^2
  (358201/228039 : -652046/228039 : 1)  C1a (30916303/859660 : -1674339/859660 : 1)
** u= 321/116 ; tau(u)= 89/205 ; 18991*x^2 - 76129*y^2 + 110962*x*z + 18991*z^2
  (-161873/1668231 : 554236/1668231 : 1)  C1a (-4589465/1644167 : 194283/1644167 : 1)
** u= 321/200 ; tau(u)= -79/121 ; 73759*x^2 - 23041*y^2 + 109282*x*z + 73759*z^2
  (-21693/393647 : -676060/393647 : 1)  C1a (11073235/875168 : -889521/875168 : 1)
** u= 321/389 ; tau(u)= 457/68 ; 93793*x^2 + 199601*y^2 + 311890*x*z + 93793*z^2
  (-116069/345941 : -13252/345941 : 1)  C1b (657387544/41999063 : -29336187/41999063 : 1)
** u= 323/178 ; tau(u)= -33/145 ; 62279*x^2 - 40961*y^2 + 105418*x*z + 62279*z^2
  (-184681/32153 : -195314/32153 : 1)  C1a (-320999/296305 : -15711/296305 : 1)
** u= 325/68 ; tau(u)= 189/257 ; -26473*x^2 - 96377*y^2 + 141346*x*z - 26473*z^2
  (6869/12349 : 8340/12349 : 1)  C1a (35507047600/154377341 : -1502253221/154377341 : 1)
** u= 325/212 ; tau(u)= -99/113 ; 80087*x^2 - 15737*y^2 + 115426*x*z + 80087*z^2
  (21953/76933 : -212040/76933 : 1)  C1a (-169803/223640 : -15313/223640 : 1)
** u= -327/349 ; tau(u)= 1025/676 ; -807023*x^2 + 136673*y^2 + 1157554*x*z - 807023*z^2
  (-37331/30675 : 153244/30675 : 1)  C2b (-3617072/1412427 : -479453/1412427 : 1)
** u= 328/337 ; tau(u)= 346/9 ; 107422*x^2 + 119554*y^2 + 227300*x*z + 107422*z^2
  (-694/817 : 4905/18791 : 1)  C1b (210175/83823 : 288835/1927929 : 1)
** u= 329/16 ; tau(u)= 297/313 ; -87697*x^2 - 107729*y^2 + 196450*x*z - 87697*z^2
  (119161/156209 : 50208/156209 : 1)  C1a (-65993993/37183391 : 4204201/37183391 : 1)
** u= 330/41 ; tau(u)= 248/289 ; -58142*x^2 - 105538*y^2 + 170404*x*z - 58142*z^2
  (2511/1042 : 391/1042 : 1)  C1a (261067/127712 : -849/9824 : 1)
** u= -332/325 ; tau(u)= 982/657 ; -753074*x^2 + 101026*y^2 + 1074548*x*z - 753074*z^2
  (-7184/10337 : -44397/10337 : 1)  C2b (-385209764/110537091 : 53519819/110537091 : 1)
** u= -336/289 ; tau(u)= 914/625 ; -668354*x^2 + 54146*y^2 + 948292*x*z - 668354*z^2
  (20229/29171 : -72250/29171 : 1)  C2b (2139207325/67880901 : 306094933/67880901 : 1)
** u= -336/317 ; tau(u)= 970/653 ; -739922*x^2 + 88082*y^2 + 1053796*x*z - 739922*z^2
  (28329/22328 : 57983/22328 : 1)  C2b (58922659/2842976 : 6888849/2842976 : 1)
** u= 337/164 ; tau(u)= 9/173 ; 53711*x^2 - 59777*y^2 + 113650*x*z + 53711*z^2
  (-77/111 : -280/2553 : 1)  C1a (210175/83823 : 288835/1927929 : 1)
** u= 339/26 ; tau(u)= 287/313 ; -81017*x^2 - 113569*y^2 + 197290*x*z - 81017*z^2
  (11/21 : 206/7077 : 1)  C1a (11315/2404 : -163965/810148 : 1)
** u= 345/64 ; tau(u)= 217/281 ; -38897*x^2 - 110833*y^2 + 166114*x*z - 38897*z^2
  (16097/58641 : 10816/58641 : 1)  C1a (-13938053/2436152 : -629253/2436152 : 1)
** u= 345/377 ; tau(u)= 409/32 ; 116977*x^2 + 165233*y^2 + 286306*x*z + 116977*z^2
  (-84036795/150900371 : -29202856/150900371 : 1)  C1b (45544044719/4766173320 : -2194923137/4766173320 : 1)
** u= 346/9 ; tau(u)= 328/337 ; -107422*x^2 - 119554*y^2 + 227300*x*z - 107422*z^2
  (3677/3023 : 20232/69529 : 1)  C1a (38800/210967 : 216935/4852241 : 1)
** u= 346/117 ; tau(u)= 112/229 ; 14834*x^2 - 92338*y^2 + 132260*x*z + 14834*z^2
  (-59131/10250321 : 4001454/10250321 : 1)  C1a (-108110391/4271303 : -4507373/4271303 : 1)
** u= 347/82 ; tau(u)= 183/265 ; -20041*x^2 - 106961*y^2 + 153898*x*z - 20041*z^2
  (1662619/11748357 : 1313578/11748357 : 1)  C1a (19423900/906213 : -810563/906213 : 1)
** u= 353/148 ; tau(u)= 57/205 ; 40559*x^2 - 80801*y^2 + 127858*x*z + 40559*z^2
  (-2123/5949 : -1436/41643 : 1)  C1a (-748441/178311 : -219283/1248177 : 1)
** u= -353/257 ; tau(u)= 867/610 ; -619591*x^2 + 7489*y^2 + 876298*x*z - 619591*z^2
  (-967/1409 : 20026/1409 : 1)  C2b (584132/110843 : 193569/110843 : 1)
** u= -356/377 ; tau(u)= 1110/733 ; -947842*x^2 + 157522*y^2 + 1358836*x*z - 947842*z^2
  (28938/99373 : 199129/99373 : 1)  C2b (-123140/124099 : 23391/124099 : 1)
** u= 357/212 ; tau(u)= -67/145 ; 85399*x^2 - 37561*y^2 + 131938*x*z + 85399*z^2
  (-108881/106403 : 109544/106403 : 1)  C1a (-359/1904 : -111/1904 : 1)
** u= 357/377 ; tau(u)= 397/20 ; 126649*x^2 + 156809*y^2 + 285058*x*z + 126649*z^2
  (-142945/198817 : -56876/198817 : 1)  C1b (-9910259/13318139 : 593133/13318139 : 1)
** u= 358/41 ; tau(u)= 276/317 ; -72814*x^2 - 124802*y^2 + 204340*x*z - 72814*z^2
  (717/304 : 55/304 : 1)  C1a (202677355/56877781 : -8500185/56877781 : 1)
** u= 359/250 ; tau(u)= -141/109 ; 105119*x^2 - 3881*y^2 + 148762*x*z + 105119*z^2
  (8231/11861 : 96890/11861 : 1)  C1a (-1671604/1532831 : -267117/1532831 : 1)
** u= -360/269 ; tau(u)= 898/629 ; -661682*x^2 + 15122*y^2 + 936004*x*z - 661682*z^2
  (-12617/6679 : -118884/6679 : 1)  C2b (85509376/28138111 : -18862073/28138111 : 1)
** u= 361/212 ; tau(u)= -63/149 ; 85919*x^2 - 40433*y^2 + 134290*x*z + 85919*z^2
  (41539/163899 : -288724/163899 : 1)  C1a (-957736/1184047 : 59371/1184047 : 1)
** u= 362/121 ; tau(u)= 120/241 ; 14882*x^2 - 101762*y^2 + 145444*x*z + 14882*z^2
  (97677/1632419 : 786764/1632419 : 1)  C1a (-16131305/1593896 : 670757/1593896 : 1)
** u= 363/365 ; tau(u)= 367/2 ; 131761*x^2 + 134681*y^2 + 266458*x*z + 131761*z^2
  (-139833/141151 : -20702/141151 : 1)  C1b (-377202924/204099481 : -16147079/204099481 : 1)
** u= -364/365 ; tau(u)= 1094/729 ; -930386*x^2 + 133954*y^2 + 1329332*x*z - 930386*z^2
  (-46585/784468 : -2156841/784468 : 1)  C2b (2913772109/190111549 : -306645001/190111549 : 1)
** u= 367/2 ; tau(u)= 363/365 ; -131761*x^2 - 134681*y^2 + 266458*x*z - 131761*z^2
  (35419/30625 : -22/625 : 1)  C1a (2898028/888297 : 126743/888297 : 1)
** u= -369/265 ; tau(u)= 899/634 ; -667751*x^2 + 4289*y^2 + 944362*x*z - 667751*z^2
  (-33335/48489 : 135146/6927 : 1)  C2b (124332/244885 : 93343/244885 : 1)
** u= 371/178 ; tau(u)= 15/193 ; 63143*x^2 - 74273*y^2 + 137866*x*z + 63143*z^2
  (-1803/3871 : -27134/65807 : 1)  C1a (5339956/720845 : -4620961/12254365 : 1)
** u= -372/325 ; tau(u)= 1022/697 ; -833234*x^2 + 72866*y^2 + 1182868*x*z - 833234*z^2
  (2772/9167 : -25225/9167 : 1)  C2b (156799595/1048964 : -21985051/1048964 : 1)
** u= 374/193 ; tau(u)= -12/181 ; 74354*x^2 - 65378*y^2 + 140020*x*z + 74354*z^2
  (-12422/216403 : -218353/216403 : 1)  C1a (-2660429460/7888859 : 135916225/7888859 : 1)
** u= -376/281 ; tau(u)= 938/657 ; -721922*x^2 + 16546*y^2 + 1021220*x*z - 721922*z^2
  (-1568/173111 : -67695/10183 : 1)  C2b (-238321/63567 : -78781/63567 : 1)
** u= 381/397 ; tau(u)= 413/16 ; 144649*x^2 + 170057*y^2 + 315730*x*z + 144649*z^2
  (-72383/68111 : 27400/68111 : 1)  C1b (-20256308336/12896483 : 964756449/12896483 : 1)
** u= 382/65 ; tau(u)= 252/317 ; -55054*x^2 - 137474*y^2 + 209428*x*z - 55054*z^2
  (16088/54155 : -7009/54155 : 1)  C1a (-354716/11997005 : 521963/11997005 : 1)
** u= 385/116 ; tau(u)= 153/269 ; 3503*x^2 - 121313*y^2 + 171634*x*z + 3503*z^2
  (161/37 : -96/37 : 1)  C1a (-3536904/6567341 : -307561/6567341 : 1)
** u= 385/272 ; tau(u)= -159/113 ; 122687*x^2 - 257*y^2 + 173506*x*z + 122687*z^2
  (7955/6171 : 285536/6171 : 1)  C1a (-43288/84067 : -55963/84067 : 1)
** u= -387/293 ; tau(u)= 973/680 ; -775031*x^2 + 21929*y^2 + 1096498*x*z - 775031*z^2
  (547/3697 : -19812/3697 : 1)  C2b (-11832864/1822549 : -3259673/1822549 : 1)
** u= 389/208 ; tau(u)= -27/181 ; 85799*x^2 - 64793*y^2 + 152050*x*z + 85799*z^2
  (136759/7275287 : 8511720/7275287 : 1)  C1a (-29524717/6312675 : 283667/1262535 : 1)
** u= 390/157 ; tau(u)= 76/233 ; 43522*x^2 - 102802*y^2 + 157876*x*z + 43522*z^2
  (201/284 : 2609/1988 : 1)  C1a (12711113/1153903 : -3966111/8077321 : 1)
** u= -392/337 ; tau(u)= 1066/729 ; -909218*x^2 + 73474*y^2 + 1290020*x*z - 909218*z^2
  (7013/2869 : 18900/2869 : 1)  C2b (1489168/3809735 : -86597/761947 : 1)
** u= 393/80 ; tau(u)= 233/313 ; -41489*x^2 - 141649*y^2 + 208738*x*z - 41489*z^2
  (1407/1223 : -1232/1223 : 1)  C1a (-775205/615611 : 45903/615611 : 1)
** u= 393/397 ; tau(u)= 401/4 ; 154417*x^2 + 160769*y^2 + 315250*x*z + 154417*z^2
  (-27603/29879 : 37000/209153 : 1)  C1b (6031559/2372712 : 2565403/16608984 : 1)
** u= -396/361 ; tau(u)= 1118/757 ; -989282*x^2 + 103826*y^2 + 1406740*x*z - 989282*z^2
  (-125276/1133797 : 540645/161971 : 1)  C2b (5765636/2513103 : 563621/2513103 : 1)
** u= 397/20 ; tau(u)= 357/377 ; -126649*x^2 - 156809*y^2 + 285058*x*z - 126649*z^2
  (39989/56801 : -15224/56801 : 1)  C1a (-187/1200 : -61/1200 : 1)
** u= 397/212 ; tau(u)= -27/185 ; 89159*x^2 - 67721*y^2 + 158338*x*z + 89159*z^2
  (28127/47555 : -84536/47555 : 1)  C1a (-1656695/527073 : -76343/527073 : 1)
** u= 399/82 ; tau(u)= 235/317 ; -41777*x^2 - 145753*y^2 + 214426*x*z - 41777*z^2
  (378255/775913 : -467122/775913 : 1)  C1a (-14634092620/51054931 : -621035973/51054931 : 1)
394
>

ここからは、 "A^4+B^4+C^4=3362*D^4の整点" と同様なので、最終的に得られた(1)の整点のみを記述する。
ここで、対応する整点が見つかった各有理数uについて、0 <= A <= B <=C を満たすように、A,B,Cを交換して、Dの小さい順に(1)の等式を並べ替えると、以下のようになる。

  • u=28/289のとき
    142571270733302981744282295959082199001^4+786691192822151378043201915694226771625^4+1034104924741982320289399202136619956564^4=334562*46221044110918601188100587643201923789^4
1426894060245285709161690433032467977291260144301763190280987846606125220883622194110748652672783789033418373807277748048369869082046597069900900122732496986704206910334531366322899991151942016154090874248006044551486656607382175287584736176665326267629082195837496189845592128855405877686283300937859706256075110524429169613740382750019402103625^4+89100558442483472581973426201462483548887086849083765998022893178919661600488545334458419983595017610075964082080471966606664764916795831875944593662074339992081347085582764000186214063771886449494669835404578337083535026655144662544050309982938812270818026850974758017948134632603018309097323605024242826170115229845032928477160730890102810917924^4+144610679725110255515898937517961196894736099704458374023592096405639801927492294791796614194818348220489611394735376903751925803357659613997470258072884706376860915641139610932834008156197691398132203559110168153603323667587826038989352277096440774118980288905598384621108215625669108581065572297127794583886277914053677765626243446661955051576991^4=334562*6218691078854168925554872006445199271580385008735486907278580687133623922821162486057487347888332133180234668322180489930612350630861062664882720940117760499700753025509284964775980386785962285470121826805171248111820230748234634993731492990913923376459816679654580077346241221776353817144931557397568995216609524926683188231885581684838246474601^4
...
  • u=180/349のとき
    250511292386383765589384700^4+1819068915189037695659468527^4+4001564747091854493865453583^4=334562*168132876585228667166089901^4
6588211923603446327453653306589640537351760296001841358873900509395711547654047110688898277523505451478198724610778664514551442100555251179448682958664989594905421910952139134446407602839323352644187820991325039117828758543397237844885056711934447659605^4+11953417744272826554146945492134545558806019103039042899924081171662565463326820396882939846242697138749440320508582388741981324271335042292497830487121042785805618111450104468368862012994194702892351607536825646424794466932074673842742838520436807816979^4+12682867214080232884698197841979531856397097974106439068929764322705226783399489700007152727816488952619956739690781236160175492317900592253293524418277952433765225245851495441815939416060175143494925992477054327011661054437160857123593358347123388347404^4=334562*616005391591789685802987514217449655662002215670528243423879764749827108029510541522161097075202647853729265212971372787983503363299398136710993223727626496942640765778049190481182600678605312399413947541975880672716598680082473681939211381979744198107^4
...



    • [参考文献]

    Last Update: 2026.02.15
    H.Nakao

    Homeに戻る[Homeに戻る]  一覧に戻る[一覧に戻る]