Integer Points on A^4+B^4+C^4=293378*D^4
[2025.12.24]A^4+B^4+C^4=293378*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。
293378=2*383^2であるので、以下では、n=383とする。
■n=393のとき、２次曲線(**)は、有理点を持たないことが確認できる。
{MAGMAでの計算]
> P2 := ProjectiveSpace(Rationals(), 2);
> N:=383;
> C := Conic(P2,-N*y^2+x^2+x*z+z^2);
> HasRationalPoint(C);
false
>
■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次曲線(3a+)と２つの２次曲線の和集合(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407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(383,1,400);
** u= 1/41 ; tau(u)= 81/40 ; -3199*x^2 + 3361*y^2 + 6562*x*z - 3199*z^2
(887/9805 : -8676/9805 : 1) C1b (-17438286/1358693 : -915787/1358693 : 1)
** u= -1/49 ; tau(u)= 99/50 ; -4999*x^2 + 4801*y^2 + 9802*x*z - 4999*z^2
(-353/543 : 910/543 : 1) C2b (-4187/1014 : 19/78 : 1)
** 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 (191619/434231 : 19231/434231 : 1)
** u= -1/241 ; tau(u)= 483/242 ; -117127*x^2 + 116161*y^2 + 233290*x*z - 117127*z^2
(35319/32023 : 4510/32023 : 1) C2b (616418/1286663 : 56997/1286663 : 1)
** u= -3/17 ; tau(u)= 37/20 ; -791*x^2 + 569*y^2 + 1378*x*z - 791*z^2
(141/95 : 88/95 : 1) C2b (-1983022/349 : -111513/349 : 1)
** u= 3/37 ; tau(u)= 71/34 ; -2303*x^2 + 2729*y^2 + 5050*x*z - 2303*z^2
(5053/7821 : -190/7821 : 1) C1b (-91106/55589 : -6219/55589 : 1)
** u= -3/101 ; tau(u)= 205/104 ; -21623*x^2 + 20393*y^2 + 42034*x*z - 21623*z^2
(20951/17421 : 5908/17421 : 1) C2b (-319181/546527 : 39507/546527 : 1)
** u= 4/5 ; tau(u)= 6 ; 14*x^2 + 34*y^2 + 52*x*z + 14*z^2
(-1/2 : -1/2 : 1) C1b (234/121 : 13/121 : 1)
** u= 4/29 ; tau(u)= 54/25 ; -1234*x^2 + 1666*y^2 + 2932*x*z - 1234*z^2
(2 : -3/7 : 1) C1b (3581/1263 : -1093/8841 : 1)
** u= -4/81 ; tau(u)= 166/85 ; -14434*x^2 + 13106*y^2 + 27572*x*z - 14434*z^2
(112/157 : 63/157 : 1) C2b (-7562/364203 : -19297/364203 : 1)
** u= -5/117 ; tau(u)= 239/122 ; -29743*x^2 + 27353*y^2 + 57146*x*z - 29743*z^2
(-45383/464953 : 530466/464953 : 1) C2b (561089/793677 : 36227/793677 : 1)
** u= -5/173 ; tau(u)= 351/178 ; -63343*x^2 + 59833*y^2 + 123226*x*z - 63343*z^2
(16243/11791 : 5662/11791 : 1) C2b (-1024549/444058 : -67897/444058 : 1)
** u= -5/269 ; tau(u)= 543/274 ; -150127*x^2 + 144697*y^2 + 294874*x*z - 150127*z^2
(13225/19761 : -51458/138327 : 1) C2b (137761/729238 : -241791/5104666 : 1)
** u= 6 ; tau(u)= 4/5 ; -14*x^2 - 34*y^2 + 52*x*z - 14*z^2
(1/2 : -1/2 : 1) C1a (2811/1882 : -131/1882 : 1)
** u= -7/45 ; tau(u)= 97/52 ; -5359*x^2 + 4001*y^2 + 9458*x*z - 5359*z^2
(4051/74305 : -81888/74305 : 1) C2b (459873/221309 : -21163/221309 : 1)
** u= -7/73 ; tau(u)= 153/80 ; -12751*x^2 + 10609*y^2 + 23458*x*z - 12751*z^2
(45/67 : 3512/6901 : 1) C2b (23811/29102 : 142229/2997506 : 1)
** u= -7/225 ; tau(u)= 457/232 ; -107599*x^2 + 101201*y^2 + 208898*x*z - 107599*z^2
(617069/686399 : -177420/686399 : 1) C2b (-1354034/562719 : -88933/562719 : 1)
** u= -8/9 ; tau(u)= 26/17 ; -514*x^2 + 98*y^2 + 740*x*z - 514*z^2
(-1/2 : -45/14 : 1) C2b (1317/623 : 689/4361 : 1)
** u= 8/25 ; tau(u)= 42/17 ; -514*x^2 + 1186*y^2 + 1828*x*z - 514*z^2
(-151/14 : 115/14 : 1) C1b (-11501/16417 : -1011/16417 : 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 (186091/102049 : 8871/102049 : 1)
** u= -8/53 ; tau(u)= 114/61 ; -7378*x^2 + 5554*y^2 + 13060*x*z - 7378*z^2
(212/279 : 5/9 : 1) C2b (244718/232005 : 2467/46401 : 1)
** u= -8/101 ; tau(u)= 210/109 ; -23698*x^2 + 20338*y^2 + 44164*x*z - 23698*z^2
(401/2841 : 2668/2841 : 1) C2b (-458606/177367 : 30543/177367 : 1)
** u= 8/333 ; tau(u)= 658/325 ; -211186*x^2 + 221714*y^2 + 433028*x*z - 211186*z^2
(463/3664 : -3111/3664 : 1) C1b (2563654/110269 : 126829/110269 : 1)
** u= -11/153 ; tau(u)= 317/164 ; -53671*x^2 + 46697*y^2 + 100610*x*z - 53671*z^2
(803/58661 : -434580/410627 : 1) C2b (172839/102934 : 54419/720538 : 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 (612593/1317682 : -58981/1317682 : 1)
** u= -11/245 ; tau(u)= 501/256 ; -130951*x^2 + 119929*y^2 + 251122*x*z - 130951*z^2
(2595/1181 : 1568/1181 : 1) C2b (-21570782/4965597 : -1282459/4965597 : 1)
** u= 12/49 ; tau(u)= 86/37 ; -2594*x^2 + 4658*y^2 + 7540*x*z - 2594*z^2
(3036/1153 : -455/1153 : 1) C1b (774347/528502 : -2733/40654 : 1)
** u= 12/109 ; tau(u)= 206/97 ; -18674*x^2 + 23618*y^2 + 42580*x*z - 18674*z^2
(-67/12586 : 78815/88102 : 1) C1b (1802663/12022 : 610401/84154 : 1)
** u= 13/53 ; tau(u)= 93/40 ; -3031*x^2 + 5449*y^2 + 8818*x*z - 3031*z^2
(-1707/4423 : -4972/4423 : 1) C1b (6331886/154107 : 288677/154107 : 1)
** u= 15/113 ; tau(u)= 211/98 ; -18983*x^2 + 25313*y^2 + 44746*x*z - 18983*z^2
(78759773/1076725509 : 850981054/1076725509 : 1) C1b (3196282/1932803 : -141777/1932803 : 1)
** u= 16/41 ; tau(u)= 66/25 ; -994*x^2 + 3106*y^2 + 4612*x*z - 994*z^2
(-451/1316 : -175/188 : 1) C1b (-1339/4038 : -199/4038 : 1)
** u= 16/257 ; tau(u)= 498/241 ; -115906*x^2 + 131842*y^2 + 248260*x*z - 115906*z^2
(-161481/112261 : 261034/112261 : 1) C1b (33749870/2659289 : 1615995/2659289 : 1)
** u= -17/81 ; tau(u)= 179/98 ; -18919*x^2 + 12833*y^2 + 32330*x*z - 18919*z^2
(-18059/2029 : 24066/2029 : 1) C2b (3696041/2281085 : 34403/456217 : 1)
** u= 17/145 ; tau(u)= 273/128 ; -32479*x^2 + 41761*y^2 + 74818*x*z - 32479*z^2
(-1/17 : -16/17 : 1) C1b (-1257926/395801 : -71263/395801 : 1)
** u= -17/257 ; tau(u)= 531/274 ; -149863*x^2 + 131809*y^2 + 282250*x*z - 149863*z^2
(3509/7013 : -4150/7013 : 1) C2b (5596401/2837861 : 250841/2837861 : 1)
** u= 17/261 ; tau(u)= 505/244 ; -118783*x^2 + 135953*y^2 + 255314*x*z - 118783*z^2
(29701/376435 : -7402332/8658005 : 1) C1b (-19786374/39545861 : 58993381/909554803 : 1)
** u= 17/313 ; tau(u)= 609/296 ; -174943*x^2 + 195649*y^2 + 371170*x*z - 174943*z^2
(-3722701/11204673 : 14275220/11204673 : 1) C1b (472759/4142534 : 196169/4142534 : 1)
** u= 17/373 ; tau(u)= 729/356 ; -253183*x^2 + 277969*y^2 + 531730*x*z - 253183*z^2
(464197/15478331 : 14306220/15478331 : 1) C1b (-1172847/5585890 : -62269/1117178 : 1)
** u= -19/49 ; tau(u)= 117/68 ; -8887*x^2 + 4441*y^2 + 14050*x*z - 8887*z^2
(153/223 : 196/223 : 1) C2b (800922/646397 : 40867/646397 : 1)
** u= -19/181 ; tau(u)= 381/200 ; -79639*x^2 + 65161*y^2 + 145522*x*z - 79639*z^2
(-129741/141361 : -293180/141361 : 1) C2b (-42433/690531 : 38587/690531 : 1)
** u= -19/225 ; tau(u)= 469/244 ; -118711*x^2 + 100889*y^2 + 220322*x*z - 118711*z^2
(40943/14993 : 29940/14993 : 1) C2b (-20163922/2064971 : 1139831/2064971 : 1)
** u= -20/29 ; tau(u)= 78/49 ; -4402*x^2 + 1282*y^2 + 6484*x*z - 4402*z^2
(5/2 : 7/2 : 1) C2b (510114/580441 : 35891/580441 : 1)
** u= -20/101 ; tau(u)= 222/121 ; -28882*x^2 + 20002*y^2 + 49684*x*z - 28882*z^2
(2896/69913 : -81037/69913 : 1) C2b (33034094/1700271 : 1826549/1700271 : 1)
** u= -20/149 ; tau(u)= 318/169 ; -56722*x^2 + 44002*y^2 + 101524*x*z - 56722*z^2
(3018/11399 : 69953/79793 : 1) C2b (-419591/147226 : -198423/1030582 : 1)
** u= 21/61 ; tau(u)= 101/40 ; -2759*x^2 + 7001*y^2 + 10642*x*z - 2759*z^2
(-987/26695 : -17924/26695 : 1) C1b (1091858/177421 : -47211/177421 : 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 (8638883/1976767 : 390219/1976767 : 1)
** u= 21/349 ; tau(u)= 677/328 ; -214727*x^2 + 243161*y^2 + 458770*x*z - 214727*z^2
(73117/503 : 68204/503 : 1) C1b (60843773/9192577 : 2834901/9192577 : 1)
** u= -23/29 ; tau(u)= 81/52 ; -4879*x^2 + 1153*y^2 + 7090*x*z - 4879*z^2
(-1691/4607 : -720/271 : 1) C2b (714618/40057 : 61613/40057 : 1)
** u= -23/41 ; tau(u)= 105/64 ; -7663*x^2 + 2833*y^2 + 11554*x*z - 7663*z^2
(9657/8327 : -10576/8327 : 1) C2b (-44588922/3314263 : -3418721/3314263 : 1)
** u= -24/25 ; tau(u)= 74/49 ; -4226*x^2 + 674*y^2 + 6052*x*z - 4226*z^2
(3/4 : 7/4 : 1) C2b (1238326/81917 : 128037/81917 : 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 (592834/17231 : -25671/17231 : 1)
** u= -24/61 ; tau(u)= 146/85 ; -13874*x^2 + 6866*y^2 + 21892*x*z - 13874*z^2
(-17/59 : -104/59 : 1) C2b (-7931158/345511 : 526107/345511 : 1)
** u= 25/89 ; tau(u)= 153/64 ; -7567*x^2 + 15217*y^2 + 24034*x*z - 7567*z^2
(829/6323 : -3456/6323 : 1) C1b (5046/7879 : -359/7879 : 1)
** u= 26/17 ; tau(u)= -8/9 ; 514*x^2 - 98*y^2 + 740*x*z + 514*z^2
(-257/113 : -3084/791 : 1) C1a (-3037/1767 : -1537/12369 : 1)
** u= -28/25 ; tau(u)= 78/53 ; -4834*x^2 + 466*y^2 + 6868*x*z - 4834*z^2
(-21/1924 : 6245/1924 : 1) C2b (61538/25589 : 6543/25589 : 1)
** u= 28/65 ; tau(u)= 102/37 ; -1954*x^2 + 7666*y^2 + 11188*x*z - 1954*z^2
(-100/207 : -209/207 : 1) C1b (-41593/96494 : 4869/96494 : 1)
** u= 28/153 ; tau(u)= 278/125 ; -30466*x^2 + 46034*y^2 + 78068*x*z - 30466*z^2
(14806/46759 : 20445/46759 : 1) C1b (455214/437243 : 23197/437243 : 1)
** u= 28/197 ; tau(u)= 366/169 ; -56338*x^2 + 76834*y^2 + 134740*x*z - 56338*z^2
(1380316/3397353 : -1279265/3397353 : 1) C1b (20259522/7890817 : 880997/7890817 : 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 (-253234/579071 : -36093/579071 : 1)
** u= 29/81 ; tau(u)= 133/52 ; -4567*x^2 + 12281*y^2 + 18530*x*z - 4567*z^2
(17477/67823 : -5976/67823 : 1) C1b (-17632158/3506965 : 166931/701393 : 1)
** u= 29/97 ; tau(u)= 165/68 ; -8407*x^2 + 17977*y^2 + 28066*x*z - 8407*z^2
(11/393 : 256/393 : 1) C1b (-6823858/190911 : 310759/190911 : 1)
** u= -29/197 ; tau(u)= 423/226 ; -101311*x^2 + 76777*y^2 + 179770*x*z - 101311*z^2
(-810247/82473 : 1015738/82473 : 1) C2b (-739702/406467 : 56687/406467 : 1)
** u= 31/65 ; tau(u)= 99/34 ; -1351*x^2 + 7489*y^2 + 10762*x*z - 1351*z^2
(1553/18425 : -4534/18425 : 1) C1b (46337/48294 : 2711/48294 : 1)
** u= -31/229 ; tau(u)= 489/260 ; -134239*x^2 + 103921*y^2 + 240082*x*z - 134239*z^2
(120229/262639 : -186616/262639 : 1) C2b (1179987469/82272209 : 62388661/82272209 : 1)
** u= 31/369 ; tau(u)= 707/338 ; -227527*x^2 + 271361*y^2 + 500810*x*z - 227527*z^2
(2214317/1390661 : 222690/1390661 : 1) C1b (21503621/4299534 : -977867/4299534 : 1)
** u= -32/53 ; tau(u)= 138/85 ; -13426*x^2 + 4594*y^2 + 20068*x*z - 13426*z^2
(-6585/28567 : -8248/4081 : 1) C2b (16121/15451 : 971/15451 : 1)
** u= 32/117 ; tau(u)= 202/85 ; -13426*x^2 + 26354*y^2 + 41828*x*z - 13426*z^2
(18127/1005415 : 697296/1005415 : 1) C1b (-1233051/83527 : 57461/83527 : 1)
** u= 33/65 ; tau(u)= 97/32 ; -959*x^2 + 7361*y^2 + 10498*x*z - 959*z^2
(-2585/48737 : -22136/48737 : 1) C1b (-2032126/2513341 : 145557/2513341 : 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 (25930061/817897 : -1188969/817897 : 1)
** u= 35/61 ; tau(u)= 87/26 ; -127*x^2 + 6217*y^2 + 8794*x*z - 127*z^2
(-83/93 : 106/93 : 1) C1b (6976153/148597 : -299757/148597 : 1)
** u= 37/20 ; tau(u)= -3/17 ; 791*x^2 - 569*y^2 + 1378*x*z + 791*z^2
(613/2037 : 436/291 : 1) C1a (-120967/29606 : -6003/29606 : 1)
** u= -37/109 ; tau(u)= 255/146 ; -41263*x^2 + 22393*y^2 + 66394*x*z - 41263*z^2
(-129/1111 : 11566/7777 : 1) C2b (-9816449/4809611 : 5813281/33667277 : 1)
** u= -37/205 ; tau(u)= 447/242 ; -115759*x^2 + 82681*y^2 + 201178*x*z - 115759*z^2
(1129705/77677 : 1257674/77677 : 1) C2b (1343831/893354 : -62313/893354 : 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 (-41753/38027 : -2163/38027 : 1)
** u= -40/49 ; tau(u)= 138/89 ; -14242*x^2 + 3202*y^2 + 20644*x*z - 14242*z^2
(173/1461 : 2828/1461 : 1) C2b (289042/102611 : -21151/102611 : 1)
** u= -40/97 ; tau(u)= 234/137 ; -35938*x^2 + 17218*y^2 + 56356*x*z - 35938*z^2
(-2995/14101 : -23916/14101 : 1) C2b (-197306/105653 : 18101/105653 : 1)
** u= -40/313 ; tau(u)= 666/353 ; -247618*x^2 + 194338*y^2 + 445156*x*z - 247618*z^2
(-238053/274403 : -563716/274403 : 1) C2b (25332393/15039059 : -1152359/15039059 : 1)
** u= -41/89 ; tau(u)= 219/130 ; -32119*x^2 + 14161*y^2 + 49642*x*z - 32119*z^2
(79/87 : 10118/10353 : 1) C2b (-8738/1609 : 79341/191471 : 1)
** u= 41/361 ; tau(u)= 681/320 ; -203119*x^2 + 258961*y^2 + 465442*x*z - 203119*z^2
(7775/17921 : -6992/17921 : 1) C1b (10615498/8051933 : 491417/8051933 : 1)
** u= 42/17 ; tau(u)= 8/25 ; 514*x^2 - 1186*y^2 + 1828*x*z + 514*z^2
(-157/721 : 248/721 : 1) C1a (26558/20971 : -1713/20971 : 1)
** u= -43/85 ; tau(u)= 213/128 ; -30919*x^2 + 12601*y^2 + 47218*x*z - 30919*z^2
(-5899/16283 : 33104/16283 : 1) C2b (-887903/133093 : 68559/133093 : 1)
** u= 43/101 ; tau(u)= 159/58 ; -4879*x^2 + 18553*y^2 + 27130*x*z - 4879*z^2
(76871/10871 : 2722/1553 : 1) C1b (-2114498/252033 : -94987/252033 : 1)
** u= -43/361 ; tau(u)= 765/404 ; -324583*x^2 + 258793*y^2 + 587074*x*z - 324583*z^2
(2135789/11073199 : 10288728/11073199 : 1) C2b (44816886/58454687 : 2736539/58454687 : 1)
** u= 44/49 ; tau(u)= 54/5 ; 1886*x^2 + 2866*y^2 + 4852*x*z + 1886*z^2
(-1580/2879 : 777/2879 : 1) C1b (7039386/13040381 : -801919/13040381 : 1)
** u= 44/349 ; tau(u)= 654/305 ; -184114*x^2 + 241666*y^2 + 429652*x*z - 184114*z^2
(-4358/5107 : 8597/5107 : 1) C1b (344293/10335223 : 491037/10335223 : 1)
** u= -48/305 ; tau(u)= 658/353 ; -246914*x^2 + 183746*y^2 + 435268*x*z - 246914*z^2
(-3642/9041 : -14341/9041 : 1) C2b (-39015610931/501671 : -2169684657/501671 : 1)
** u= 49/261 ; tau(u)= 473/212 ; -87487*x^2 + 133841*y^2 + 226130*x*z - 87487*z^2
(-191101/327473 : 446880/327473 : 1) C1b (8330930/2970369 : 360985/2970369 : 1)
** u= 53/157 ; tau(u)= 261/104 ; -18823*x^2 + 46489*y^2 + 70930*x*z - 18823*z^2
(113573/28287 : 3628/4041 : 1) C1b (126465691/10698431 : 5536609/10698431 : 1)
** u= 54/5 ; tau(u)= 44/49 ; -1886*x^2 - 2866*y^2 + 4852*x*z - 1886*z^2
(1580/2879 : -777/2879 : 1) C1a (2682358/117321 : 124063/117321 : 1)
** u= 54/25 ; tau(u)= 4/29 ; 1234*x^2 - 1666*y^2 + 2932*x*z + 1234*z^2
(-83/152 : -25/1064 : 1) C1a (37834/35347 : -20701/247429 : 1)
** u= -55/109 ; tau(u)= 273/164 ; -50767*x^2 + 20737*y^2 + 77554*x*z - 50767*z^2
(-91167/3659 : -147064/3659 : 1) C2b (1097981/285513 : -65423/285513 : 1)
** u= -55/153 ; tau(u)= 361/208 ; -83503*x^2 + 43793*y^2 + 133346*x*z - 83503*z^2
(-33275/14731 : 63384/14731 : 1) C2b (4218/69541 : -4223/69541 : 1)
** u= 55/353 ; tau(u)= 651/298 ; -174583*x^2 + 246193*y^2 + 426826*x*z - 174583*z^2
(82751/571839 : 393322/571839 : 1) C1b (-28282474/491673 : 1355257/491673 : 1)
** u= -56/41 ; tau(u)= 138/97 ; -15682*x^2 + 226*y^2 + 22180*x*z - 15682*z^2
(-1327/146 : 11945/146 : 1) C2b (9959/19706 : 5187/19706 : 1)
** u= 56/61 ; tau(u)= 66/5 ; 3086*x^2 + 4306*y^2 + 7492*x*z + 3086*z^2
(-87/128 : -47/128 : 1) C1b (27914/43987 : -2909/43987 : 1)
** u= 56/81 ; tau(u)= 106/25 ; 1886*x^2 + 9986*y^2 + 14372*x*z + 1886*z^2
(-463/113 : 180/113 : 1) C1b (-220207/35603 : 9467/35603 : 1)
** u= 56/85 ; tau(u)= 114/29 ; 1454*x^2 + 11314*y^2 + 16132*x*z + 1454*z^2
(-57907/340229 : -113068/340229 : 1) C1b (70144454/670913 : -3029067/670913 : 1)
** u= 56/289 ; tau(u)= 522/233 ; -105442*x^2 + 163906*y^2 + 275620*x*z - 105442*z^2
(9303/49508 : -29291/49508 : 1) C1b (802517/295135 : 6949/59027 : 1)
** u= 57/97 ; tau(u)= 137/40 ; 49*x^2 + 15569*y^2 + 22018*x*z + 49*z^2
(-4611/113023 : 26396/113023 : 1) C1b (53917829/335113 : -2316813/335113 : 1)
** u= 57/193 ; tau(u)= 329/136 ; -33743*x^2 + 71249*y^2 + 111490*x*z - 33743*z^2
(139247/443397 : -75356/443397 : 1) C1b (34760273/2629642 : 1539279/2629642 : 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 (38027/41753 : 2163/41753 : 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 (-181277887/44475653 : -12419649/44475653 : 1)
** u= 61/65 ; tau(u)= 69/4 ; 3689*x^2 + 4729*y^2 + 8482*x*z + 3689*z^2
(-9063/8731 : 4288/8731 : 1) C1b (-437829014/16371113 : -20853941/16371113 : 1)
** u= -61/85 ; tau(u)= 231/146 ; -38911*x^2 + 10729*y^2 + 57082*x*z - 38911*z^2
(30037/59109 : 80606/59109 : 1) C2b (-72353/43163 : 8881/43163 : 1)
** u= 61/225 ; tau(u)= 389/164 ; -50071*x^2 + 97529*y^2 + 155042*x*z - 50071*z^2
(14335/292681 : 193428/292681 : 1) C1b (5121111/9999814 : -439667/9999814 : 1)
** u= -64/73 ; tau(u)= 210/137 ; -33442*x^2 + 6562*y^2 + 48196*x*z - 33442*z^2
(1187/38511 : -85028/38511 : 1) C2b (891626/44503 : -84343/44503 : 1)
** u= -64/109 ; tau(u)= 282/173 ; -55762*x^2 + 19666*y^2 + 83620*x*z - 55762*z^2
(13451/76439 : 112732/76439 : 1) C2b (-379153/392853 : -52511/392853 : 1)
** u= -64/121 ; tau(u)= 306/185 ; -64354*x^2 + 25186*y^2 + 97732*x*z - 64354*z^2
(-2515/28307 : 338602/198149 : 1) C2b (65669/81447 : -31123/570129 : 1)
** u= 66/5 ; tau(u)= 56/61 ; -3086*x^2 - 4306*y^2 + 7492*x*z - 3086*z^2
(3501/2107 : -932/2107 : 1) C1a (7022/6623 : -353/6623 : 1)
** u= 66/25 ; tau(u)= 16/41 ; 994*x^2 - 3106*y^2 + 4612*x*z + 994*z^2
(1015/2603 : -2534/2603 : 1) C1a (-306667/53122 : -13197/53122 : 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 (-377378/4495 : -4475/899 : 1)
** u= -67/61 ; tau(u)= 189/128 ; -28279*x^2 + 2953*y^2 + 40210*x*z - 28279*z^2
(-271/511 : 2256/511 : 1) C2b (40070/397411 : -49375/397411 : 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 (106383/806926 : -50281/806926 : 1)
** u= -68/113 ; tau(u)= 294/181 ; -60898*x^2 + 20914*y^2 + 91060*x*z - 60898*z^2
(-94426/1327371 : 2387903/1327371 : 1) C2b (2318970/1375457 : -130655/1375457 : 1)
** u= 68/265 ; tau(u)= 462/197 ; -72994*x^2 + 135826*y^2 + 218068*x*z - 72994*z^2
(246538/1422907 : 746677/1422907 : 1) C1b (1283717/332434 : 55559/332434 : 1)
** u= 69/4 ; tau(u)= 61/65 ; -3689*x^2 - 4729*y^2 + 8482*x*z - 3689*z^2
(565/333 : -44/333 : 1) C1a (-13163323/3023342 : 712871/3023342 : 1)
** u= 69/185 ; tau(u)= 301/116 ; -22151*x^2 + 63689*y^2 + 95362*x*z - 22151*z^2
(-13883/125951 : -90568/125951 : 1) C1b (30010498/6154097 : -1291041/6154097 : 1)
** u= 71/34 ; tau(u)= 3/37 ; 2303*x^2 - 2729*y^2 + 5050*x*z + 2303*z^2
(599/14357 : 1970/2051 : 1) C1a (18322/125977 : -6651/125977 : 1)
** u= 73/221 ; tau(u)= 369/148 ; -38479*x^2 + 92353*y^2 + 141490*x*z - 38479*z^2
(459/25369 : -688/1103 : 1) C1b (739478235/6511462 : -32993075/6511462 : 1)
** u= 74/49 ; tau(u)= -24/25 ; 4226*x^2 - 674*y^2 + 6052*x*z + 4226*z^2
(-461/1401 : 2800/1401 : 1) C1a (-118/257 : 21/257 : 1)
** u= -76/169 ; tau(u)= 414/245 ; -114274*x^2 + 51346*y^2 + 177172*x*z - 114274*z^2
(7240/3743 : -7371/3743 : 1) C2b (-479707531/117384079 : 37760881/117384079 : 1)
** u= 76/197 ; tau(u)= 318/121 ; -23506*x^2 + 71842*y^2 + 106900*x*z - 23506*z^2
(-81883/626328 : 454817/626328 : 1) C1b (5328481/1689475 : 46047/337895 : 1)
** u= -76/241 ; tau(u)= 558/317 ; -195202*x^2 + 110386*y^2 + 317140*x*z - 195202*z^2
(34258/103811 : -104475/103811 : 1) C2b (137507/220782 : -10657/220782 : 1)
** u= 76/269 ; tau(u)= 462/193 ; -68722*x^2 + 138946*y^2 + 219220*x*z - 68722*z^2
(150023/483998 : 111499/483998 : 1) C1b (8917146/91801 : 403409/91801 : 1)
** u= 77/97 ; tau(u)= 117/20 ; 5129*x^2 + 12889*y^2 + 19618*x*z + 5129*z^2
(-289/465 : 292/465 : 1) C1b (-153355282/25465863 : -6630377/25465863 : 1)
** u= -77/197 ; tau(u)= 471/274 ; -144223*x^2 + 71689*y^2 + 227770*x*z - 144223*z^2
(22227/54923 : -56426/54923 : 1) C2b (11340506/3221495 : -124029/644299 : 1)
** u= 78/49 ; tau(u)= -20/29 ; 4402*x^2 - 1282*y^2 + 6484*x*z + 4402*z^2
(-2713/2880 : -3773/2880 : 1) C1a (-5017/968763 : 78443/968763 : 1)
** u= 78/53 ; tau(u)= -28/25 ; 4834*x^2 - 466*y^2 + 6868*x*z + 4834*z^2
(-1877/1242 : 4265/1242 : 1) C1a (-110731/12609 : 14183/12609 : 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 (-22463/66762 : -6577/66762 : 1)
** u= 79/153 ; tau(u)= 227/74 ; -4711*x^2 + 40577*y^2 + 57770*x*z - 4711*z^2
(-2209603/59070449 : 24320850/59070449 : 1) C1b (4642569/63565 : 39989/12713 : 1)
** u= 80/81 ; tau(u)= 82 ; 6398*x^2 + 6722*y^2 + 13124*x*z + 6398*z^2
(-2227/2651 : -342/2651 : 1) C1b (-25206/53731 : 2369/53731 : 1)
** u= 80/117 ; tau(u)= 154/37 ; 3662*x^2 + 20978*y^2 + 30116*x*z + 3662*z^2
(-2803/10199 : -4638/10199 : 1) C1b (-1971709/45522 : 85141/45522 : 1)
** u= -80/117 ; tau(u)= 314/197 ; -71218*x^2 + 20978*y^2 + 104996*x*z - 71218*z^2
(1369/727 : -1782/727 : 1) C2b (7735726/216297 : -613579/216297 : 1)
** u= -80/149 ; tau(u)= 378/229 ; -98482*x^2 + 38002*y^2 + 149284*x*z - 98482*z^2
(56954/352381 : -501357/352381 : 1) C2b (120791/173157 : 9331/173157 : 1)
** u= 80/261 ; tau(u)= 442/181 ; -59122*x^2 + 129842*y^2 + 201764*x*z - 59122*z^2
(-13679/5167 : -14394/5167 : 1) C1b (2455318/4012357 : -182143/4012357 : 1)
** u= 81/40 ; tau(u)= 1/41 ; 3199*x^2 - 3361*y^2 + 6562*x*z + 3199*z^2
(-115/89 : 12/89 : 1) C1a (-25206/53731 : 2369/53731 : 1)
** u= 81/52 ; tau(u)= -23/29 ; 4879*x^2 - 1153*y^2 + 7090*x*z + 4879*z^2
(-2087/7201 : -12060/7201 : 1) C1a (-4181/2281 : 277/2281 : 1)
** u= 82 ; tau(u)= 80/81 ; -6398*x^2 - 6722*y^2 + 13124*x*z - 6398*z^2
(2227/2651 : -342/2651 : 1) C1a (-17438286/1358693 : -915787/1358693 : 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 (-211959125/1111279 : 126135/15223 : 1)
** u= 85/101 ; tau(u)= 117/16 ; 6713*x^2 + 13177*y^2 + 20914*x*z + 6713*z^2
(-1269/2717 : -944/2717 : 1) C1b (-741794/188459 : 32059/188459 : 1)
** u= 86/37 ; tau(u)= 12/49 ; 2594*x^2 - 4658*y^2 + 7540*x*z + 2594*z^2
(-1679/76 : 1169/76 : 1) C1a (-849058/537773 : -38349/537773 : 1)
** u= 87/26 ; tau(u)= 35/61 ; 127*x^2 - 6217*y^2 + 8794*x*z + 127*z^2
(319/12585 : -2986/12585 : 1) C1a (-7687514/9434927 : 519273/9434927 : 1)
** u= 88/89 ; tau(u)= 90 ; 7742*x^2 + 8098*y^2 + 15844*x*z + 7742*z^2
(-823/680 : 73/680 : 1) C1b (-179583/52286 : 8119/52286 : 1)
** u= -88/205 ; tau(u)= 498/293 ; -163954*x^2 + 76306*y^2 + 255748*x*z - 163954*z^2
(-4301/25791 : 42904/25791 : 1) C2b (5551589074702/25946764541 : 366124942313/25946764541 : 1)
** u= 89/97 ; tau(u)= 105/8 ; 7793*x^2 + 10897*y^2 + 18946*x*z + 7793*z^2
(-23805/18809 : 10964/18809 : 1) C1b (-3347641/199102 : 155827/199102 : 1)
** u= 89/169 ; tau(u)= 249/80 ; -4879*x^2 + 49201*y^2 + 69922*x*z - 4879*z^2
(-43957/1843461 : -672568/1843461 : 1) C1b (1151362/617397 : 54613/617397 : 1)
** u= -89/281 ; tau(u)= 651/370 ; -265879*x^2 + 150001*y^2 + 431722*x*z - 265879*z^2
(76745/79809 : -64042/79809 : 1) C2b (9116461/586582 : -537417/586582 : 1)
** u= 90 ; tau(u)= 88/89 ; -7742*x^2 - 8098*y^2 + 15844*x*z - 7742*z^2
(1783/2152 : -201/2152 : 1) C1a (-1091402/45459 : -56317/45459 : 1)
** u= -91/153 ; tau(u)= 397/244 ; -110791*x^2 + 38537*y^2 + 165890*x*z - 110791*z^2
(-409/521 : 1476/521 : 1) C2b (139084334/544319 : 10399207/544319 : 1)
** u= -91/205 ; tau(u)= 501/296 ; -166951*x^2 + 75769*y^2 + 259282*x*z - 166951*z^2
(11/61305 : -90988/61305 : 1) C2b (-33601187/493369 : 2267223/493369 : 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 (178389034/6863413 : 7670727/6863413 : 1)
** u= 93/40 ; tau(u)= 13/53 ; 3031*x^2 - 5449*y^2 + 8818*x*z + 3031*z^2
(-203/1185 : -644/1185 : 1) C1a (354702/78469 : 17867/78469 : 1)
** u= 93/337 ; tau(u)= 581/244 ; -110423*x^2 + 218489*y^2 + 346210*x*z - 110423*z^2
(2413/9863 : 276968/719999 : 1) C1b (156830/417163 : -18015/417163 : 1)
** u= -95/233 ; tau(u)= 561/328 ; -206143*x^2 + 99553*y^2 + 323746*x*z - 206143*z^2
(84481/211779 : 222412/211779 : 1) C2b (208976483/6306567 : 13358377/6306567 : 1)
** u= 97/32 ; tau(u)= 33/65 ; 959*x^2 - 7361*y^2 + 10498*x*z + 959*z^2
(-19653/947555 : 42968/135365 : 1) C1a (-361451/319543 : -19851/319543 : 1)
** u= 97/52 ; tau(u)= -7/45 ; 5359*x^2 - 4001*y^2 + 9458*x*z + 5359*z^2
(-107/173 : -108/173 : 1) C1a (1187798/591069 : 88811/591069 : 1)
** u= -97/153 ; tau(u)= 403/250 ; -115591*x^2 + 37409*y^2 + 171818*x*z - 115591*z^2
(-86399/4872529 : -8678490/4872529 : 1) C2b (496718/1661681 : 106661/1661681 : 1)
** u= -97/361 ; tau(u)= 819/458 ; -410119*x^2 + 251233*y^2 + 680170*x*z - 410119*z^2
(1092607/807227 : -790362/807227 : 1) C2b (-533757119509/50956583319 : 33645695611/50956583319 : 1)
** u= 99/34 ; tau(u)= 31/65 ; 1351*x^2 - 7489*y^2 + 10762*x*z + 1351*z^2
(333/2783 : 1658/2783 : 1) C1a (-17165166/658909 : -740437/658909 : 1)
** u= 99/50 ; tau(u)= -1/49 ; 4999*x^2 - 4801*y^2 + 9802*x*z + 4999*z^2
(-4759/5571 : -1330/5571 : 1) C1a (-151417/29406 : -19/78 : 1)
** u= 100/153 ; tau(u)= 206/53 ; 4382*x^2 + 36818*y^2 + 52436*x*z + 4382*z^2
(-4189/18056 : -8175/18056 : 1) C1b (6018279/1165061 : 268411/1165061 : 1)
** u= 100/377 ; tau(u)= 654/277 ; -143458*x^2 + 274258*y^2 + 437716*x*z - 143458*z^2
(-13002/618287 : 461389/618287 : 1) C1b (101674/20423321 : 930731/20423321 : 1)
** u= 101/40 ; tau(u)= 21/61 ; 2759*x^2 - 7001*y^2 + 10642*x*z + 2759*z^2
(-4345/27411 : 11068/27411 : 1) C1a (-407642/1089937 : -47223/1089937 : 1)
** u= 101/121 ; tau(u)= 141/20 ; 9401*x^2 + 19081*y^2 + 30082*x*z + 9401*z^2
(-33991/86547 : 1144/5091 : 1) C1b (-84284621/628519 : 3814911/628519 : 1)
** u= 102/37 ; tau(u)= 28/65 ; 1954*x^2 - 7666*y^2 + 11188*x*z + 1954*z^2
(-4/3757 : 1891/3757 : 1) C1a (1802/1507 : 111/1507 : 1)
** u= 103/225 ; tau(u)= 347/122 ; -19159*x^2 + 90641*y^2 + 131018*x*z - 19159*z^2
(31873/919813 : -369690/919813 : 1) C1b (8942967/29291419 : -1274909/29291419 : 1)
** u= -104/81 ; tau(u)= 266/185 ; -57634*x^2 + 2306*y^2 + 81572*x*z - 57634*z^2
(103/346 : 1413/346 : 1) C2b (-39389/79746 : 23881/79746 : 1)
** u= 104/193 ; tau(u)= 282/89 ; -5026*x^2 + 63682*y^2 + 90340*x*z - 5026*z^2
(2246/79391 : 15649/79391 : 1) C1b (-4067205/1945681 : 198185/1945681 : 1)
** u= 105/8 ; tau(u)= 89/97 ; -7793*x^2 - 10897*y^2 + 18946*x*z - 7793*z^2
(207/211 : 116/211 : 1) C1a (277703/109986 : -12061/109986 : 1)
** u= 105/64 ; tau(u)= -23/41 ; 7663*x^2 - 2833*y^2 + 11554*x*z + 7663*z^2
(2309/661 : 4672/661 : 1) C1a (152302/467481 : -42307/467481 : 1)
** u= 106/25 ; tau(u)= 56/81 ; -1886*x^2 - 9986*y^2 + 14372*x*z - 1886*z^2
(161/262 : 207/262 : 1) C1a (3241067/141894 : 139813/141894 : 1)
** u= -107/377 ; tau(u)= 861/484 ; -457063*x^2 + 272809*y^2 + 752770*x*z - 457063*z^2
(53617/125287 : -112156/125287 : 1) C2b (534107318/50652907 : 30307019/50652907 : 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 (-116261/790159 : 370737/5531113 : 1)
** u= -111/305 ; tau(u)= 721/416 ; -333791*x^2 + 173729*y^2 + 532162*x*z - 333791*z^2
(27345/87031 : -93272/87031 : 1) C2b (335299/3216638 : 190827/3216638 : 1)
** u= 112/197 ; tau(u)= 282/85 ; -1906*x^2 + 65074*y^2 + 92068*x*z - 1906*z^2
(-3529/9965 : 7282/9965 : 1) C1b (12469186/201703 : 535779/201703 : 1)
** u= -112/277 ; tau(u)= 666/389 ; -290098*x^2 + 140914*y^2 + 456100*x*z - 290098*z^2
(-4061/6714 : 14663/6714 : 1) C2b (-638530902/47499467 : 43536031/47499467 : 1)
** u= -112/289 ; tau(u)= 690/401 ; -309058*x^2 + 154498*y^2 + 488644*x*z - 309058*z^2
(6429/3743 : 5882/3743 : 1) C2b (-5375617/4476389 : 572481/4476389 : 1)
** u= 114/29 ; tau(u)= 56/85 ; -1454*x^2 - 11314*y^2 + 16132*x*z - 1454*z^2
(219/2320 : 163/2320 : 1) C1a (-46961/56787 : -3317/56787 : 1)
** u= 114/61 ; tau(u)= -8/53 ; 7378*x^2 - 5554*y^2 + 13060*x*z + 7378*z^2
(-7819/4723 : -4900/4723 : 1) C1a (-17377/35195 : -323/7039 : 1)
** u= -116/305 ; tau(u)= 726/421 ; -341026*x^2 + 172594*y^2 + 540532*x*z - 341026*z^2
(22288/68407 : 73843/68407 : 1) C2b (-1373749/6848737 : -498023/6848737 : 1)
** u= -116/317 ; tau(u)= 750/433 ; -361522*x^2 + 187522*y^2 + 575956*x*z - 361522*z^2
(-35796/150701 : 250645/150701 : 1) C2b (46163/381513 : -22439/381513 : 1)
** u= 117/16 ; tau(u)= 85/101 ; -6713*x^2 - 13177*y^2 + 20914*x*z - 6713*z^2
(1269/2717 : -944/2717 : 1) C1a (-1233051/83527 : 57461/83527 : 1)
** u= 117/20 ; tau(u)= 77/97 ; -5129*x^2 - 12889*y^2 + 19618*x*z - 5129*z^2
(10239/33971 : 232/1477 : 1) C1a (3213631/948258 : 138353/948258 : 1)
** u= 117/68 ; tau(u)= -19/49 ; 8887*x^2 - 4441*y^2 + 14050*x*z + 8887*z^2
(-3097/1263 : -3164/1263 : 1) C1a (-1110775/29414 : 70205/29414 : 1)
** u= -119/137 ; tau(u)= 393/256 ; -116911*x^2 + 23377*y^2 + 168610*x*z - 116911*z^2
(-7611/660443 : 1489280/660443 : 1) C2b (-5204505/485699 : -538775/485699 : 1)
** u= -119/365 ; tau(u)= 849/484 ; -454351*x^2 + 252289*y^2 + 734962*x*z - 454351*z^2
(114063/792205 : 943624/792205 : 1) C2b (343152478/522926791 : 25377729/522926791 : 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 (-113475027/5080394 : 5513653/5080394 : 1)
** u= 123/317 ; tau(u)= 511/194 ; -60143*x^2 + 185849*y^2 + 276250*x*z - 60143*z^2
(154593/1168237 : 425374/1168237 : 1) C1b (4256539/3880874 : -224559/3880874 : 1)
** u= -123/397 ; tau(u)= 917/520 ; -525671*x^2 + 300089*y^2 + 856018*x*z - 525671*z^2
(56067/17585 : -56884/17585 : 1) C2b (12883474/8298077 : 620949/8298077 : 1)
** u= -124/337 ; tau(u)= 798/461 ; -409666*x^2 + 211762*y^2 + 652180*x*z - 409666*z^2
(43213/32736 : 36455/32736 : 1) C2b (82214527/8047030 : -981729/1609406 : 1)
** u= -124/369 ; tau(u)= 862/493 ; -470722*x^2 + 256946*y^2 + 758420*x*z - 470722*z^2
(54656/742163 : -945945/742163 : 1) C2b (-333113882/55909227 : 22993829/55909227 : 1)
** u= 128/153 ; tau(u)= 178/25 ; 15134*x^2 + 30434*y^2 + 48068*x*z + 15134*z^2
(-3635/9949 : -1152/9949 : 1) C1b (-37062643/580646 : -1674833/580646 : 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 (436408474/28080763 : 21025661/28080763 : 1)
** u= -129/353 ; tau(u)= 835/482 ; -448007*x^2 + 232577*y^2 + 713866*x*z - 448007*z^2
(70625/485013 : 597998/485013 : 1) C2b (2821318/5003849 : -247431/5003849 : 1)
** u= -132/157 ; tau(u)= 446/289 ; -149618*x^2 + 31874*y^2 + 216340*x*z - 149618*z^2
(820842/718661 : -1258255/718661 : 1) C2b (458099/853129 : -59319/853129 : 1)
** u= 132/245 ; tau(u)= 358/113 ; -8114*x^2 + 102626*y^2 + 145588*x*z - 8114*z^2
(29/666 : -2023/15318 : 1) C1b (55843/14369 : -56301/330487 : 1)
** u= -132/361 ; tau(u)= 854/493 ; -468674*x^2 + 243218*y^2 + 746740*x*z - 468674*z^2
(75104/239017 : 256595/239017 : 1) C2b (-807724522/7661683 : -51442881/7661683 : 1)
** u= 133/52 ; tau(u)= 29/81 ; 4567*x^2 - 12281*y^2 + 18530*x*z + 4567*z^2
(-37/4267 : -2556/4267 : 1) C1a (-9809946/615461 : -429469/615461 : 1)
** u= -133/349 ; tau(u)= 831/482 ; -446959*x^2 + 225913*y^2 + 708250*x*z - 446959*z^2
(-19589/87523 : 145910/87523 : 1) C2b (24948842/683187 : -1568879/683187 : 1)
** u= -136/121 ; tau(u)= 378/257 ; -113602*x^2 + 10786*y^2 + 161380*x*z - 113602*z^2
(-373/15104 : -49885/15104 : 1) C2b (29982/178177 : -22153/178177 : 1)
** u= -136/145 ; tau(u)= 426/281 ; -139426*x^2 + 23554*y^2 + 199972*x*z - 139426*z^2
(-14561/1023 : -37252/1023 : 1) C2b (2654482/32871 : 277057/32871 : 1)
** u= 137/40 ; tau(u)= 57/97 ; -49*x^2 - 15569*y^2 + 22018*x*z - 49*z^2
(2551/3427 : -3508/3427 : 1) C1a (-124359394/1232453 : 5343843/1232453 : 1)
** u= 138/85 ; tau(u)= -32/53 ; 13426*x^2 - 4594*y^2 + 20068*x*z + 13426*z^2
(-17961/8711 : 21934/8711 : 1) C1a (-170442/14819 : -419/511 : 1)
** u= 138/89 ; tau(u)= -40/49 ; 14242*x^2 - 3202*y^2 + 20644*x*z + 14242*z^2
(-101/710 : 1351/710 : 1) C1a (6229143/1912099 : 704047/1912099 : 1)
** u= 138/97 ; tau(u)= -56/41 ; 15682*x^2 - 226*y^2 + 22180*x*z + 15682*z^2
(-144/121 : -863/121 : 1) C1a (-1105/934 : -285/934 : 1)
** u= 140/289 ; tau(u)= 438/149 ; -24802*x^2 + 147442*y^2 + 211444*x*z - 24802*z^2
(-25616/206893 : 122111/206893 : 1) C1b (1031193/446939 : -46501/446939 : 1)
** u= 140/333 ; tau(u)= 526/193 ; -54898*x^2 + 202178*y^2 + 296276*x*z - 54898*z^2
(-179282/326641 : -351447/326641 : 1) C1b (5165206/486699 : -222959/486699 : 1)
** u= 141/20 ; tau(u)= 101/121 ; -9401*x^2 - 19081*y^2 + 30082*x*z - 9401*z^2
(829/1965 : -572/1965 : 1) C1a (-39618/1648693 : 75377/1648693 : 1)
** u= 141/245 ; tau(u)= 349/104 ; -1751*x^2 + 100169*y^2 + 141682*x*z - 1751*z^2
(68157/9925319 : -874804/9925319 : 1) C1b (1541519/447862 : 68751/447862 : 1)
** u= 143/193 ; tau(u)= 243/50 ; 15449*x^2 + 54049*y^2 + 79498*x*z + 15449*z^2
(-4889/20855 : 4338/20855 : 1) C1b (650646/320813 : 34151/320813 : 1)
** u= 143/225 ; tau(u)= 307/82 ; 7001*x^2 + 80801*y^2 + 114698*x*z + 7001*z^2
(-2249/3271 : -21090/22897 : 1) C1b (-1030662/125267 : 310567/876869 : 1)
** u= 143/241 ; tau(u)= 339/98 ; 1241*x^2 + 95713*y^2 + 135370*x*z + 1241*z^2
(-14913/1739 : -5810/1739 : 1) C1b (-10585/31771 : 1435/31771 : 1)
** u= 145/197 ; tau(u)= 249/52 ; 15617*x^2 + 56593*y^2 + 83026*x*z + 15617*z^2
(-20227/103455 : -1888/103455 : 1) C1b (58571/965527 : 42811/965527 : 1)
** u= 146/85 ; tau(u)= -24/61 ; 13874*x^2 - 6866*y^2 + 21892*x*z + 13874*z^2
(-1853/688 : 1957/688 : 1) C1a (216863/80291 : -17733/80291 : 1)
** u= 147/181 ; tau(u)= 215/34 ; 19297*x^2 + 43913*y^2 + 67834*x*z + 19297*z^2
(-21325/10539 : 9926/10539 : 1) C1b (-7722539/1617566 : 333483/1617566 : 1)
** u= -148/157 ; tau(u)= 462/305 ; -164146*x^2 + 27394*y^2 + 235348*x*z - 164146*z^2
(549136/1886425 : -3772333/1886425 : 1) C2b (-1790061/1193734 : 292151/1193734 : 1)
** u= -149/125 ; tau(u)= 399/274 ; -127951*x^2 + 9049*y^2 + 181402*x*z - 127951*z^2
(-32881/119579 : 544330/119579 : 1) C2b (-6039201/1054778 : 1103983/1054778 : 1)
** u= 149/181 ; tau(u)= 213/32 ; 20153*x^2 + 43321*y^2 + 67570*x*z + 20153*z^2
(-63077/63853 : -50344/63853 : 1) C1b (-9648833/2305435 : 83277/461087 : 1)
** u= -149/261 ; tau(u)= 671/410 ; -313999*x^2 + 114041*y^2 + 472442*x*z - 313999*z^2
(47225/12353 : 64374/12353 : 1) C2b (-67587222/1103599 : 5024737/1103599 : 1)
** u= 149/349 ; tau(u)= 549/200 ; -57799*x^2 + 221401*y^2 + 323602*x*z - 57799*z^2
(-2802089/4077429 : -4805140/4077429 : 1) C1b (-29583291/92057737 : 4438099/92057737 : 1)
** u= 151/369 ; tau(u)= 587/218 ; -72247*x^2 + 249521*y^2 + 367370*x*z - 72247*z^2
(-85019/4727149 : -379686/675307 : 1) C1b (-10113121/2792586 : -482617/2792586 : 1)
** u= -152/113 ; tau(u)= 378/265 ; -117346*x^2 + 2434*y^2 + 165988*x*z - 117346*z^2
(-3373/4375 : -49776/4375 : 1) C2b (-4040714/33917 : -1212827/33917 : 1)
** u= -152/221 ; tau(u)= 594/373 ; -255154*x^2 + 74578*y^2 + 375940*x*z - 255154*z^2
(-186/3161 : 42733/22127 : 1) C2b (-158261/79745 : 25021/111643 : 1)
** u= 153/64 ; tau(u)= 25/89 ; 7567*x^2 - 15217*y^2 + 24034*x*z + 7567*z^2
(-865/7391 : -4176/7391 : 1) C1a (-37062643/580646 : -1674833/580646 : 1)
** u= 153/80 ; tau(u)= -7/73 ; 12751*x^2 - 10609*y^2 + 23458*x*z + 12751*z^2
(-45/67 : -3512/6901 : 1) C1a (312954/41383 : 1866263/4262449 : 1)
** u= 154/37 ; tau(u)= 80/117 ; -3662*x^2 - 20978*y^2 + 30116*x*z - 3662*z^2
(1703/13168 : 1191/13168 : 1) C1a (-567279/51626 : 24929/51626 : 1)
** u= -156/113 ; tau(u)= 382/269 ; -120386*x^2 + 1202*y^2 + 170260*x*z - 120386*z^2
(84/131 : -931/131 : 1) C2b (3112883/593518 : 1172121/593518 : 1)
** u= 156/205 ; tau(u)= 254/49 ; 19534*x^2 + 59714*y^2 + 88852*x*z + 19534*z^2
(-15871/39636 : -18431/39636 : 1) C1b (-1393699607/9824147 : -61331991/9824147 : 1)
** u= 156/265 ; tau(u)= 374/109 ; 574*x^2 + 116114*y^2 + 164212*x*z + 574*z^2
(-5143/418326 : 46663/418326 : 1) C1b (-14132857/3553117 : -625641/3553117 : 1)
** u= -156/365 ; tau(u)= 886/521 ; -518546*x^2 + 242114*y^2 + 809332*x*z - 518546*z^2
(-22835/833372 : -1245869/833372 : 1) C2b (903898/968897 : -52581/968897 : 1)
** u= 157/257 ; tau(u)= 357/100 ; 4649*x^2 + 107449*y^2 + 152098*x*z + 4649*z^2
(-1321/261 : 640/261 : 1) C1b (2990266/785513 : -133901/785513 : 1)
** u= 159/58 ; tau(u)= 43/101 ; 4879*x^2 - 18553*y^2 + 27130*x*z + 4879*z^2
(-183/1111 : 190/1111 : 1) C1a (3094606/1260211 : 154891/1260211 : 1)
** u= 159/377 ; tau(u)= 595/218 ; -69767*x^2 + 258977*y^2 + 379306*x*z - 69767*z^2
(729/6685 : -2246/6685 : 1) C1b (-32538058/11305069 : -1589793/11305069 : 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 (-4061277/202478 : 194507/202478 : 1)
** u= 165/68 ; tau(u)= 29/97 ; 8407*x^2 - 17977*y^2 + 28066*x*z + 8407*z^2
(-531/5897 : -3392/5897 : 1) C1a (1967181/139118 : 90889/139118 : 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 (-1787234/407477 : -2321427/6927109 : 1)
** u= 166/85 ; tau(u)= -4/81 ; 14434*x^2 - 13106*y^2 + 27572*x*z + 14434*z^2
(5875/922232 : -973719/922232 : 1) C1a (-1887226/99997 : -96263/99997 : 1)
** u= -167/125 ; tau(u)= 417/292 ; -142639*x^2 + 3361*y^2 + 201778*x*z - 142639*z^2
(13053/73843 : -60740/10549 : 1) C2b (13085459/327047 : -3599199/327047 : 1)
** u= -168/277 ; tau(u)= 722/445 ; -367826*x^2 + 125234*y^2 + 549508*x*z - 367826*z^2
(178162/560577 : 760247/560577 : 1) C2b (7689737/3261499 : 455331/3261499 : 1)
** u= -169/361 ; tau(u)= 891/530 ; -533239*x^2 + 232081*y^2 + 822442*x*z - 533239*z^2
(30707/242139 : 332462/242139 : 1) C2b (1973709734/47721213 : 132164737/47721213 : 1)
** u= 172/257 ; tau(u)= 342/85 ; 15134*x^2 + 102514*y^2 + 146548*x*z + 15134*z^2
(-18866/55169 : 31401/55169 : 1) C1b (-1524994/720843 : -69901/720843 : 1)
** u= 172/353 ; tau(u)= 534/181 ; -35938*x^2 + 219634*y^2 + 314740*x*z - 35938*z^2
(-11801/235594 : 114407/235594 : 1) C1b (10827794/1480901 : -465387/1480901 : 1)
** u= -173/317 ; tau(u)= 807/490 ; -450271*x^2 + 171049*y^2 + 681178*x*z - 450271*z^2
(579987/236311 : -697634/236311 : 1) C2b (-243623722/15027153 : -18300383/15027153 : 1)
** u= 173/369 ; tau(u)= 565/196 ; -46903*x^2 + 242393*y^2 + 349154*x*z - 46903*z^2
(-9857/406981 : 194544/406981 : 1) C1b (12805718/21392151 : -1015147/21392151 : 1)
** u= -175/313 ; tau(u)= 801/488 ; -445663*x^2 + 165313*y^2 + 672226*x*z - 445663*z^2
(31917/90343 : 114124/90343 : 1) C2b (117632393/19168861 : -7715953/19168861 : 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 (-16302114/826421 : -816893/826421 : 1)
** u= -177/193 ; tau(u)= 563/370 ; -242471*x^2 + 43169*y^2 + 348298*x*z - 242471*z^2
(191/1303 : 19466/9121 : 1) C2b (-1441106/316457 : 1204683/2215199 : 1)
** u= 178/25 ; tau(u)= 128/153 ; -15134*x^2 - 30434*y^2 + 48068*x*z - 15134*z^2
(194/103 : 87/103 : 1) C1a (5046/7879 : -359/7879 : 1)
** u= 179/98 ; tau(u)= -17/81 ; 18919*x^2 - 12833*y^2 + 32330*x*z + 18919*z^2
(-4189/3469 : 2646/3469 : 1) C1a (-944529198/663554105 : -8876407/132710821 : 1)
** u= 181/221 ; tau(u)= 261/40 ; 29561*x^2 + 64921*y^2 + 100882*x*z + 29561*z^2
(-52141/155697 : 18412/155697 : 1) C1b (-12580733/9265502 : -597203/9265502 : 1)
** u= -184/205 ; tau(u)= 594/389 ; -268786*x^2 + 50194*y^2 + 386692*x*z - 268786*z^2
(-32239/96862 : -282603/96862 : 1) C2b (-215868397/104340474 : 29884403/104340474 : 1)
** u= 184/369 ; tau(u)= 554/185 ; -34594*x^2 + 238466*y^2 + 340772*x*z - 34594*z^2
(38621/4483277 : 1633596/4483277 : 1) C1b (3472368891/701008721 : 149961481/701008721 : 1)
** u= -185/369 ; tau(u)= 923/554 ; -579607*x^2 + 238097*y^2 + 886154*x*z - 579607*z^2
(-7387/3245 : 15738/3245 : 1) C2b (194584711/49556419 : -11602201/49556419 : 1)
** u= -187/205 ; tau(u)= 597/392 ; -272359*x^2 + 49081*y^2 + 391378*x*z - 272359*z^2
(-5223/158299 : -381836/158299 : 1) C2b (352157/1668538 : 147511/1668538 : 1)
** u= 189/128 ; tau(u)= -67/61 ; 28279*x^2 - 2953*y^2 + 40210*x*z + 28279*z^2
(1487/8383 : 29392/8383 : 1) C1a (15362030/270561 : 2073715/270561 : 1)
** u= -191/233 ; tau(u)= 657/424 ; -323071*x^2 + 72097*y^2 + 468130*x*z - 323071*z^2
(63613/90007 : -131372/90007 : 1) C2b (-16494583382/1208646933 : -122881253/92972841 : 1)
** u= 191/265 ; tau(u)= 339/74 ; 25529*x^2 + 103969*y^2 + 151402*x*z + 25529*z^2
(-21793/6351 : -8662/6351 : 1) C1b (96722/383171 : -17907/383171 : 1)
** u= 191/289 ; tau(u)= 387/98 ; 17273*x^2 + 130561*y^2 + 186250*x*z + 17273*z^2
(-782521/6885499 : -1154538/6885499 : 1) C1b (-483733/347983 : 24559/347983 : 1)
** u= 192/205 ; tau(u)= 218/13 ; 36526*x^2 + 47186*y^2 + 84388*x*z + 36526*z^2
(-7713/13289 : 746/13289 : 1) C1b (-8370062/745931 : -390189/745931 : 1)
** u= 192/289 ; tau(u)= 386/97 ; 18046*x^2 + 130178*y^2 + 185860*x*z + 18046*z^2
(-27139/47049 : -37604/47049 : 1) C1b (-9728843/3628958 : 433803/3628958 : 1)
** u= 192/373 ; tau(u)= 554/181 ; -28658*x^2 + 241394*y^2 + 343780*x*z - 28658*z^2
(740749/83577161 : 27224332/83577161 : 1) C1b (15568018/2146943 : 669933/2146943 : 1)
** u= -196/373 ; tau(u)= 942/569 ; -609106*x^2 + 239842*y^2 + 925780*x*z - 609106*z^2
(-115866/369479 : 738941/369479 : 1) C2b (-20545196195/105969202 : -1463395605/105969202 : 1)
** u= 196/389 ; tau(u)= 582/193 ; -36082*x^2 + 264226*y^2 + 377140*x*z - 36082*z^2
(-12216/110011 : -59927/110011 : 1) C1b (21220842/41102545 : -383465/8220509 : 1)
** u= -197/269 ; tau(u)= 735/466 ; -395503*x^2 + 105913*y^2 + 579034*x*z - 395503*z^2
(990435/261403 : 1582042/261403 : 1) C2b (-461365462/47591619 : -41826431/47591619 : 1)
** u= 202/85 ; tau(u)= 32/117 ; 13426*x^2 - 26354*y^2 + 41828*x*z + 13426*z^2
(-17885/382037 : -252336/382037 : 1) C1a (-741794/188459 : 32059/188459 : 1)
** u= 203/333 ; tau(u)= 463/130 ; 7409*x^2 + 180569*y^2 + 255578*x*z + 7409*z^2
(-85841/2957875 : 9678/2957875 : 1) C1b (-443436881/40103419 : -19089479/40103419 : 1)
** u= -204/193 ; tau(u)= 590/397 ; -273602*x^2 + 32882*y^2 + 389716*x*z - 273602*z^2
(1135/654 : -2341/654 : 1) C2b (130429/604439 : 64833/604439 : 1)
** u= -204/361 ; tau(u)= 926/565 ; -596834*x^2 + 219026*y^2 + 899092*x*z - 596834*z^2
(829066/7383021 : 11192881/7383021 : 1) C2b (75011/71857 : 4413/71857 : 1)
** u= 204/377 ; tau(u)= 550/173 ; -18242*x^2 + 242642*y^2 + 344116*x*z - 18242*z^2
(-12212/64707 : -38035/64707 : 1) C1b (2503/136973 : -5889/136973 : 1)
** u= 205/104 ; tau(u)= -3/101 ; 21623*x^2 - 20393*y^2 + 42034*x*z + 21623*z^2
(-18825/130859 : -115996/130859 : 1) C1a (-44665618/7747121 : -2138187/7747121 : 1)
** u= 206/53 ; tau(u)= 100/153 ; -4382*x^2 - 36818*y^2 + 52436*x*z - 4382*z^2
(12853/151022 : -5505/151022 : 1) C1a (-61269/599866 : -26219/599866 : 1)
** u= 206/97 ; tau(u)= 12/109 ; 18674*x^2 - 23618*y^2 + 42580*x*z + 18674*z^2
(-324/4397 : 25045/30779 : 1) C1a (19382/77213 : 29703/540491 : 1)
** u= 208/245 ; tau(u)= 282/37 ; 40526*x^2 + 76786*y^2 + 122788*x*z + 40526*z^2
(-131/135 : -98/135 : 1) C1b (-3602179862/18583831 : -164305839/18583831 : 1)
** u= 208/261 ; tau(u)= 314/53 ; 37646*x^2 + 92978*y^2 + 141860*x*z + 37646*z^2
(-146273/361933 : -138102/361933 : 1) C1b (-69187/92198 : 4441/92198 : 1)
** u= 208/293 ; tau(u)= 378/85 ; 28814*x^2 + 128434*y^2 + 186148*x*z + 28814*z^2
(-9527/3908 : -5493/3908 : 1) C1b (-13032121/12634957 : -725921/12634957 : 1)
** u= -208/397 ; tau(u)= 1002/605 ; -688786*x^2 + 271954*y^2 + 1047268*x*z - 688786*z^2
(43275322/7514966259 : -11907459113/7514966259 : 1) C2b (18042386/9680067 : -976037/9680067 : 1)
** u= 209/325 ; tau(u)= 441/116 ; 16769*x^2 + 167569*y^2 + 238162*x*z + 16769*z^2
(-7009/24745 : -13428/24745 : 1) C1b (-4638106/1649451 : 207277/1649451 : 1)
** u= 210/109 ; tau(u)= -8/101 ; 23698*x^2 - 20338*y^2 + 44164*x*z + 23698*z^2
(2265/3023 : 5612/3023 : 1) C1a (-470251/186466 : -21417/186466 : 1)
** u= 210/137 ; tau(u)= -64/73 ; 33442*x^2 - 6562*y^2 + 48196*x*z + 33442*z^2
(-3487/7989 : -13514/7989 : 1) C1a (203872187/14492318 : 20972821/14492318 : 1)
** u= 211/98 ; tau(u)= 15/113 ; 18983*x^2 - 25313*y^2 + 44746*x*z + 18983*z^2
(15 : 14 : 1) C1a (305327/3182 : 14733/3182 : 1)
** u= 211/269 ; tau(u)= 327/58 ; 37793*x^2 + 100201*y^2 + 151450*x*z + 37793*z^2
(-41409/23459 : 24778/23459 : 1) C1b (4761887/726587 : -221793/726587 : 1)
** u= -212/153 ; tau(u)= 518/365 ; -221506*x^2 + 1874*y^2 + 313268*x*z - 221506*z^2
(-64/41 : -1059/41 : 1) C2b (-1165962/24371 : -552791/24371 : 1)
** u= 213/32 ; tau(u)= 149/181 ; -20153*x^2 - 43321*y^2 + 67570*x*z - 20153*z^2
(109311/184631 : -100312/184631 : 1) C1a (-182195018/3819617 : -8274519/3819617 : 1)
** u= 213/128 ; tau(u)= -43/85 ; 30919*x^2 - 12601*y^2 + 47218*x*z + 30919*z^2
(3749/105045 : 169072/105045 : 1) C1a (-47631/41702 : 2623/41702 : 1)
** u= 215/34 ; tau(u)= 147/181 ; -19297*x^2 - 43913*y^2 + 67834*x*z - 19297*z^2
(61967/69269 : 53242/69269 : 1) C1a (-39816662/2194849 : 1818537/2194849 : 1)
** u= 215/337 ; tau(u)= 459/122 ; 16457*x^2 + 180913*y^2 + 256906*x*z + 16457*z^2
(-47839/3229 : -3198/3229 : 1) C1b (108541/684134 : -30119/684134 : 1)
** u= -217/265 ; tau(u)= 747/482 ; -417559*x^2 + 93361*y^2 + 605098*x*z - 417559*z^2
(6113/21481 : 37146/21481 : 1) C2b (88438542/69636733 : 5940661/69636733 : 1)
** u= 218/13 ; tau(u)= 192/205 ; -36526*x^2 - 47186*y^2 + 84388*x*z - 36526*z^2
(16983/29269 : 1604/29269 : 1) C1a (-979453/212126 : -3093/12478 : 1)
** u= 219/130 ; tau(u)= -41/89 ; 32119*x^2 - 14161*y^2 + 49642*x*z + 32119*z^2
(-95/263 : -35654/31297 : 1) C1a (-8431/2611 : -56589/310709 : 1)
** u= -220/157 ; tau(u)= 534/377 ; -235858*x^2 + 898*y^2 + 333556*x*z - 235858*z^2
(321/60268 : 973057/60268 : 1) C2b (34662/32671 : 17989/32671 : 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 (-545207314/4978127 : 47638449/4978127 : 1)
** u= 220/333 ; tau(u)= 446/113 ; 22862*x^2 + 173378*y^2 + 247316*x*z + 22862*z^2
(-64712/598135 : 86517/598135 : 1) C1b (114457717/4484098 : 4960867/4484098 : 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 (3622/24709 : -1073/24709 : 1)
** u= 222/121 ; tau(u)= -20/101 ; 28882*x^2 - 20002*y^2 + 49684*x*z + 28882*z^2
(16165/97554 : 134299/97554 : 1) C1a (-2360914/136793 : -130069/136793 : 1)
** u= -223/169 ; tau(u)= 561/392 ; -257599*x^2 + 7393*y^2 + 364450*x*z - 257599*z^2
(39033/65267 : -275548/65267 : 1) C2b (25413/185513 : -42749/185513 : 1)
** u= 223/233 ; tau(u)= 243/10 ; 49529*x^2 + 58849*y^2 + 108778*x*z + 49529*z^2
(-9619/13381 : 21402/93667 : 1) C1b (-882887/494949 : -272179/3464643 : 1)
** u= -224/233 ; tau(u)= 690/457 ; -367522*x^2 + 58402*y^2 + 526276*x*z - 367522*z^2
(17012/31157 : 56159/31157 : 1) C2b (-9541046/2163663 : -1210439/2163663 : 1)
** u= 227/74 ; tau(u)= 79/153 ; 4711*x^2 - 40577*y^2 + 57770*x*z + 4711*z^2
(47659/2165273 : 831570/2165273 : 1) C1a (-1144050042/139564727 : 49198321/139564727 : 1)
** u= -229/349 ; tau(u)= 927/578 ; -615727*x^2 + 191161*y^2 + 911770*x*z - 615727*z^2
(-109721/113767 : -374170/113767 : 1) C2b (27696742/36960933 : 2161879/36960933 : 1)
** u= 229/365 ; tau(u)= 501/136 ; 15449*x^2 + 214009*y^2 + 303442*x*z + 15449*z^2
(-128613/2374345 : 157564/2374345 : 1) C1b (-543159349/966920679 : 46665907/966920679 : 1)
** u= 231/146 ; tau(u)= -61/85 ; 38911*x^2 - 10729*y^2 + 57082*x*z + 38911*z^2
(-2693/22289 : 38842/22289 : 1) C1a (221759/271957 : 37767/271957 : 1)
** u= -232/193 ; tau(u)= 618/425 ; -307426*x^2 + 20674*y^2 + 435748*x*z - 307426*z^2
(34669/32181 : 98780/32181 : 1) C2b (569071/103581 : 83171/103581 : 1)
** u= 232/245 ; tau(u)= 258/13 ; 53486*x^2 + 66226*y^2 + 120388*x*z + 53486*z^2
(-4439/6229 : -1736/6229 : 1) C1b (-582973/3338146 : -152143/3338146 : 1)
** u= 234/137 ; tau(u)= -40/97 ; 35938*x^2 - 17218*y^2 + 56356*x*z + 35938*z^2
(-1657/385 : -284/55 : 1) C1a (-217598/158601 : -11033/158601 : 1)
** u= 235/293 ; tau(u)= 351/58 ; 48497*x^2 + 116473*y^2 + 178426*x*z + 48497*z^2
(-761/2407 : -2734/16849 : 1) C1b (-141771/539198 : 162443/3774386 : 1)
** u= 239/122 ; tau(u)= -5/117 ; 29743*x^2 - 27353*y^2 + 57146*x*z + 29743*z^2
(-20437/14389 : -8058/14389 : 1) C1a (-372483/114554 : 17119/114554 : 1)
** u= -240/181 ; tau(u)= 602/421 ; -296882*x^2 + 7922*y^2 + 420004*x*z - 296882*z^2
(25461/96379 : -492242/96379 : 1) C2b (-44226382/99859 : -11653851/99859 : 1)
** u= 243/10 ; tau(u)= 223/233 ; -49529*x^2 - 58849*y^2 + 108778*x*z - 49529*z^2
(281/303 : 818/2121 : 1) C1a (888343/1977589 : -605453/13843123 : 1)
** u= 243/50 ; tau(u)= 143/193 ; -15449*x^2 - 54049*y^2 + 79498*x*z - 15449*z^2
(14609/19379 : -15750/19379 : 1) C1a (-77447829/7956337 : -3478351/7956337 : 1)
** u= 245/313 ; tau(u)= 381/68 ; 50777*x^2 + 135913*y^2 + 205186*x*z + 50777*z^2
(-38043/86095 : 40432/86095 : 1) C1b (-19462259/6578729 : 840791/6578729 : 1)
** u= -245/389 ; tau(u)= 1023/634 ; -743887*x^2 + 242617*y^2 + 1106554*x*z - 743887*z^2
(17123/174935 : 284722/174935 : 1) C2b (2362462/11576813 : -782483/11576813 : 1)
** u= -247/205 ; tau(u)= 657/452 ; -347599*x^2 + 23041*y^2 + 492658*x*z - 347599*z^2
(-32071/519447 : 2107684/519447 : 1) C2b (-1316847/2059861 : -523151/2059861 : 1)
** u= 248/261 ; tau(u)= 274/13 ; 61166*x^2 + 74738*y^2 + 136580*x*z + 61166*z^2
(-15916/12353 : -5205/12353 : 1) C1b (-240043/50942 : 10841/50942 : 1)
** u= 249/52 ; tau(u)= 145/197 ; -15617*x^2 - 56593*y^2 + 83026*x*z - 15617*z^2
(95/471 : 44/471 : 1) C1a (3936067/610161 : 169277/610161 : 1)
** u= 249/80 ; tau(u)= 89/169 ; 4879*x^2 - 49201*y^2 + 69922*x*z + 4879*z^2
(611/15187 : -6008/15187 : 1) C1a (2107982/2196641 : -135629/2196641 : 1)
** u= -251/293 ; tau(u)= 837/544 ; -528871*x^2 + 108697*y^2 + 763570*x*z - 528871*z^2
(673227/1101853 : -1703368/1101853 : 1) C2b (27124710567/1689469358 : -2488242307/1689469358 : 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 (164798/216901 : 195317/1518307 : 1)
** u= 254/49 ; tau(u)= 156/205 ; -19534*x^2 - 59714*y^2 + 88852*x*z - 19534*z^2
(16126/67387 : -6811/67387 : 1) C1a (-1235102/1696159 : -101691/1696159 : 1)
** u= 255/146 ; tau(u)= -37/109 ; 41263*x^2 - 22393*y^2 + 66394*x*z + 41263*z^2
(-7335/3859 : 45718/27013 : 1) C1a (-3475334/1449 : -1513129/10143 : 1)
** u= 258/13 ; tau(u)= 232/245 ; -53486*x^2 - 66226*y^2 + 120388*x*z - 53486*z^2
(332480/216061 : 60053/216061 : 1) C1a (1288086/2678291 : 116987/2678291 : 1)
** u= 259/365 ; tau(u)= 471/106 ; 44609*x^2 + 199369*y^2 + 288922*x*z + 44609*z^2
(-25395/98473 : 36194/98473 : 1) C1b (11034953/5771414 : 574977/5771414 : 1)
** u= -260/197 ; tau(u)= 654/457 ; -350098*x^2 + 10018*y^2 + 495316*x*z - 350098*z^2
(1225778/104992757 : -615569431/104992757 : 1) C2b (628003734/3962203 : 158840711/3962203 : 1)
** u= 261/40 ; tau(u)= 181/221 ; -29561*x^2 - 64921*y^2 + 100882*x*z - 29561*z^2
(7687/10483 : 6948/10483 : 1) C1a (2455318/4012357 : -182143/4012357 : 1)
** u= 261/104 ; tau(u)= 53/157 ; 18823*x^2 - 46489*y^2 + 70930*x*z + 18823*z^2
(-336323/8233 : -203964/8233 : 1) C1a (-69187/92198 : 4441/92198 : 1)
** u= 264/373 ; tau(u)= 482/109 ; 45934*x^2 + 208562*y^2 + 302020*x*z + 45934*z^2
(-13913/82907 : -10672/82907 : 1) C1b (11884607/25892053 : 1308183/25892053 : 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 (-32318738/33576421 : 1726023/33576421 : 1)
** u= 265/281 ; tau(u)= 297/16 ; 69713*x^2 + 87697*y^2 + 158434*x*z + 69713*z^2
(-68483/62891 : 30144/62891 : 1) C1b (-66637/35791 : 2923/35791 : 1)
** u= 266/185 ; tau(u)= -104/81 ; 57634*x^2 - 2306*y^2 + 81572*x*z + 57634*z^2
(1015/37217 : 189684/37217 : 1) C1a (2333291/172207 : 528209/172207 : 1)
** u= -268/373 ; tau(u)= 1014/641 ; -749938*x^2 + 206434*y^2 + 1100020*x*z - 749938*z^2
(-786/7871 : 949/463 : 1) C2b (-2517046/4219805 : -103671/843961 : 1)
** u= 269/289 ; tau(u)= 309/20 ; 71561*x^2 + 94681*y^2 + 167842*x*z + 71561*z^2
(-4065/2341 : -488/2341 : 1) C1b (-3588918/3228469 : -176251/3228469 : 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 (1360154/1094403 : 102647/1094403 : 1)
** u= 273/128 ; tau(u)= 17/145 ; 32479*x^2 - 41761*y^2 + 74818*x*z + 32479*z^2
(-24275/77181 : 41648/77181 : 1) C1a (1591586/538727 : 91349/538727 : 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 (-821170/1555693 : 68805/1555693 : 1)
** u= 273/164 ; tau(u)= -55/109 ; 50767*x^2 - 20737*y^2 + 77554*x*z + 50767*z^2
(91147/976907 : -1640036/976907 : 1) C1a (-924954/1234973 : -65741/1234973 : 1)
** u= 274/13 ; tau(u)= 248/261 ; -61166*x^2 - 74738*y^2 + 136580*x*z - 61166*z^2
(3583/4562 : 1527/4562 : 1) C1a (464522/705433 : -31651/705433 : 1)
** u= 278/125 ; tau(u)= 28/153 ; 30466*x^2 - 46034*y^2 + 78068*x*z + 30466*z^2
(4445/22 : 3639/22 : 1) C1a (279668526/2307709 : -13201859/2307709 : 1)
** u= 280/293 ; tau(u)= 306/13 ; 78062*x^2 + 93298*y^2 + 172036*x*z + 78062*z^2
(-477481/308695 : 36084/308695 : 1) C1b (-398707/88926 : 18007/88926 : 1)
** u= -280/313 ; tau(u)= 906/593 ; -624898*x^2 + 117538*y^2 + 899236*x*z - 624898*z^2
(-1197/1202 : -5129/1202 : 1) C2b (-715831583/46239054 : 74824313/46239054 : 1)
** u= -280/373 ; tau(u)= 1026/653 ; -774418*x^2 + 199858*y^2 + 1131076*x*z - 774418*z^2
(12769/48365 : -4632/2845 : 1) C2b (-13760314/786837 : 1230061/786837 : 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 (1419919/677077 : -72897/677077 : 1)
** u= 282/37 ; tau(u)= 208/245 ; -40526*x^2 - 76786*y^2 + 122788*x*z - 40526*z^2
(1949/3882 : -1463/3882 : 1) C1a (26801509/323042 : -1219773/323042 : 1)
** u= 282/85 ; tau(u)= 112/197 ; 1906*x^2 - 65074*y^2 + 92068*x*z + 1906*z^2
(435/334 : -463/334 : 1) C1a (-12475182/1572121 : -538997/1572121 : 1)
** u= 282/89 ; tau(u)= 104/193 ; 5026*x^2 - 63682*y^2 + 90340*x*z + 5026*z^2
(-327/9851 : 1760/9851 : 1) C1a (-1272742/312453 : 55667/312453 : 1)
** u= 282/173 ; tau(u)= -64/109 ; 55762*x^2 - 19666*y^2 + 83620*x*z + 55762*z^2
(-1544/16709 : -3749/2387 : 1) C1a (-206937/201067 : -12413/201067 : 1)
** u= 284/313 ; tau(u)= 342/29 ; 78974*x^2 + 115282*y^2 + 197620*x*z + 78974*z^2
(-64247/128678 : 1047/128678 : 1) C1b (-2682633/2486422 : 134201/2486422 : 1)
** u= -293/333 ; tau(u)= 959/626 ; -697903*x^2 + 135929*y^2 + 1005530*x*z - 697903*z^2
(10877/187073 : 406494/187073 : 1) C2b (48553734/13676509 : -3958679/13676509 : 1)
** u= 294/181 ; tau(u)= -68/113 ; 60898*x^2 - 20914*y^2 + 91060*x*z + 60898*z^2
(7294/52593 : 99395/52593 : 1) C1a (2338274/950531 : -230459/950531 : 1)
** u= 296/369 ; tau(u)= 442/73 ; 76958*x^2 + 184706*y^2 + 282980*x*z + 76958*z^2
(-1843471/3318494 : 1835253/3318494 : 1) C1b (4269650485/39184623 : 191432095/39184623 : 1)
** u= 297/16 ; tau(u)= 265/281 ; -69713*x^2 - 87697*y^2 + 158434*x*z - 69713*z^2
(1555/1459 : -696/1459 : 1) C1a (-424246814/198352191 : -26304811/198352191 : 1)
** u= 299/349 ; tau(u)= 399/50 ; 84401*x^2 + 154201*y^2 + 248602*x*z + 84401*z^2
(-13/23 : 10/23 : 1) C1b (37042177982/29042349 : 1699834139/29042349 : 1)
** u= 301/116 ; tau(u)= 69/185 ; 22151*x^2 - 63689*y^2 + 95362*x*z + 22151*z^2
(8721/20753 : -21148/20753 : 1) C1a (1820789/1489142 : -115227/1489142 : 1)
** u= 306/13 ; tau(u)= 280/293 ; -78062*x^2 - 93298*y^2 + 172036*x*z - 78062*z^2
(140615/155033 : -59532/155033 : 1) C1a (9921677/74601 : -485143/74601 : 1)
** u= 306/185 ; tau(u)= -64/121 ; 64354*x^2 - 25186*y^2 + 97732*x*z + 64354*z^2
(-1241/4288 : 38511/30016 : 1) C1a (-384998/3081 : 190807/21567 : 1)
** u= 307/82 ; tau(u)= 143/225 ; -7001*x^2 - 80801*y^2 + 114698*x*z - 7001*z^2
(19/149 : 318/1043 : 1) C1a (58702/54731 : 23437/383117 : 1)
** u= -308/269 ; tau(u)= 846/577 ; -570994*x^2 + 49858*y^2 + 810580*x*z - 570994*z^2
(1022/2697 : 7103/2697 : 1) C2b (12358726/83451 : 1791943/83451 : 1)
** u= -308/365 ; tau(u)= 1038/673 ; -810994*x^2 + 171586*y^2 + 1172308*x*z - 810994*z^2
(15632/12609 : 23659/12609 : 1) C2b (13696961/2667643 : -1133969/2667643 : 1)
** u= -308/389 ; tau(u)= 1086/697 ; -876754*x^2 + 207778*y^2 + 1274260*x*z - 876754*z^2
(-3572883/15432896 : -37377055/15432896 : 1) C2b (26807086/19513031 : -1743227/19513031 : 1)
** u= 309/20 ; tau(u)= 269/289 ; -71561*x^2 - 94681*y^2 + 167842*x*z - 71561*z^2
(116673/154001 : 60316/154001 : 1) C1a (15173754/8863307 : -670253/8863307 : 1)
** u= -313/337 ; tau(u)= 987/650 ; -747031*x^2 + 129169*y^2 + 1072138*x*z - 747031*z^2
(-29607/60061 : -11866/3533 : 1) C2b (18058/990051 : -101759/990051 : 1)
** u= 314/53 ; tau(u)= 208/261 ; -37646*x^2 - 92978*y^2 + 141860*x*z - 37646*z^2
(313/149 : 150/149 : 1) C1a (126465691/10698431 : 5536609/10698431 : 1)
** u= 314/197 ; tau(u)= -80/117 ; 71218*x^2 - 20978*y^2 + 104996*x*z + 71218*z^2
(-395/8 : 717/8 : 1) C1a (-12953203/14903649 : -914579/14903649 : 1)
** u= 317/164 ; tau(u)= -11/153 ; 53671*x^2 - 46697*y^2 + 100610*x*z + 53671*z^2
(-10921/57727 : -357540/404089 : 1) C1a (-71438614/22063051 : -23209853/154441357 : 1)
** u= 318/121 ; tau(u)= 76/197 ; 23506*x^2 - 71842*y^2 + 106900*x*z + 23506*z^2
(83092/969219 : -655325/969219 : 1) C1a (-44837/850778 : -37113/850778 : 1)
** u= 318/169 ; tau(u)= -20/149 ; 56722*x^2 - 44002*y^2 + 101524*x*z + 56722*z^2
(5586/21745 : -213473/152215 : 1) C1a (-341522/4831409 : 1786131/33819863 : 1)
** u= -319/277 ; tau(u)= 873/596 ; -608671*x^2 + 51697*y^2 + 863890*x*z - 608671*z^2
(263 : -900 : 1) C2b (2602470/1700171 : 273485/1700171 : 1)
** u= -319/373 ; tau(u)= 1065/692 ; -855967*x^2 + 176497*y^2 + 1235986*x*z - 855967*z^2
(-157145/65041 : 460276/65041 : 1) C2b (-1990726/4246093 : 555059/4246093 : 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 (66762/22463 : -6577/22463 : 1)
** u= 327/58 ; tau(u)= 211/269 ; -37793*x^2 - 100201*y^2 + 151450*x*z - 37793*z^2
(23459/41409 : 24778/41409 : 1) C1a (13718050/1307249 : -597525/1307249 : 1)
** u= 328/389 ; tau(u)= 450/61 ; 100142*x^2 + 195058*y^2 + 310084*x*z + 100142*z^2
(-91349/72999 : -59840/72999 : 1) C1b (-37871066/17545277 : 1650653/17545277 : 1)
** u= 329/136 ; tau(u)= 57/193 ; 33743*x^2 - 71249*y^2 + 111490*x*z + 33743*z^2
(7271/471659 : -332788/471659 : 1) C1a (-7166798/4712995 : -65889/942599 : 1)
** u= -332/245 ; tau(u)= 822/577 ; -555634*x^2 + 9826*y^2 + 785908*x*z - 555634*z^2
(20/27 : -2443/459 : 1) C2b (-48734/793 : 270807/13481 : 1)
** u= 332/337 ; tau(u)= 342/5 ; 110174*x^2 + 116914*y^2 + 227188*x*z + 110174*z^2
(-2695/2102 : -47/14714 : 1) C1b (-142666989/41384398 : -45068237/289690786 : 1)
** u= 339/74 ; tau(u)= 191/265 ; -25529*x^2 - 103969*y^2 + 151402*x*z - 25529*z^2
(169915/330381 : 218762/330381 : 1) C1a (18293719/1772198 : 788289/1772198 : 1)
** u= 339/98 ; tau(u)= 143/241 ; -1241*x^2 - 95713*y^2 + 135370*x*z - 1241*z^2
(80281/327843 : -2590/4491 : 1) C1a (-86939626/25967031 : 3908623/25967031 : 1)
** u= 342/5 ; tau(u)= 332/337 ; -110174*x^2 - 116914*y^2 + 227188*x*z - 110174*z^2
(506/643 : 257/4501 : 1) C1a (127779/64231 : 39407/449617 : 1)
** u= 342/29 ; tau(u)= 284/313 ; -78974*x^2 - 115282*y^2 + 197620*x*z - 78974*z^2
(5658/11327 : -197/11327 : 1) C1a (108350570/117194123 : -5856335/117194123 : 1)
** u= 342/85 ; tau(u)= 172/257 ; -15134*x^2 - 102514*y^2 + 146548*x*z - 15134*z^2
(1439/1096 : 1331/1096 : 1) C1a (8521798/2207829 : -370639/2207829 : 1)
** u= -344/389 ; tau(u)= 1122/733 ; -956242*x^2 + 184306*y^2 + 1377220*x*z - 956242*z^2
(-39909/6818 : 14665/974 : 1) C2b (-15024865/1149922 : 1564975/1149922 : 1)
** u= 344/397 ; tau(u)= 450/53 ; 112718*x^2 + 196882*y^2 + 320836*x*z + 112718*z^2
(-301/706 : -4665/34594 : 1) C1b (-1084282/624609 : 2359499/30605841 : 1)
** u= 347/122 ; tau(u)= 103/225 ; 19159*x^2 - 90641*y^2 + 131018*x*z + 19159*z^2
(3323/1525 : 3186/1525 : 1) C1a (-1156474/1177863 : 66257/1177863 : 1)
** u= 349/104 ; tau(u)= 141/245 ; 1751*x^2 - 100169*y^2 + 141682*x*z + 1751*z^2
(169/7405 : 1652/7405 : 1) C1a (-6348613/1579802 : -280311/1579802 : 1)
** u= 351/58 ; tau(u)= 235/293 ; -48497*x^2 - 116473*y^2 + 178426*x*z - 48497*z^2
(201/107 : -746/749 : 1) C1a (112327/610054 : 184543/4270378 : 1)
** u= 351/178 ; tau(u)= -5/173 ; 63343*x^2 - 59833*y^2 + 123226*x*z + 63343*z^2
(-52665/49561 : 12694/49561 : 1) C1a (-23996186/3459607 : 1162271/3459607 : 1)
** u= 357/100 ; tau(u)= 157/257 ; -4649*x^2 - 107449*y^2 + 152098*x*z - 4649*z^2
(24393/791857 : 13652/791857 : 1) C1a (-4477818/1117121 : -199811/1117121 : 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 (-38137/145138 : -8391/145138 : 1)
** u= 358/113 ; tau(u)= 132/245 ; 8114*x^2 - 102626*y^2 + 145588*x*z + 8114*z^2
(870/1993 : -38717/45839 : 1) C1a (1449014/21977 : -1435611/505471 : 1)
** u= 361/208 ; tau(u)= -55/153 ; 83503*x^2 - 43793*y^2 + 133346*x*z + 83503*z^2
(-401591/1024625 : -146832/146375 : 1) C1a (151591/228138 : 21689/228138 : 1)
** u= 366/169 ; tau(u)= 28/197 ; 56338*x^2 - 76834*y^2 + 134740*x*z + 56338*z^2
(1/14 : 13/14 : 1) C1a (-7790648674/6162416381 : 366030013/6162416381 : 1)
** u= -368/333 ; tau(u)= 1034/701 ; -847378*x^2 + 86354*y^2 + 1204580*x*z - 847378*z^2
(-3371/92851 : 298458/92851 : 1) C2b (1460229/888070 : 28261/177614 : 1)
** u= -368/389 ; tau(u)= 1146/757 ; -1010674*x^2 + 167218*y^2 + 1448740*x*z - 1010674*z^2
(-363091/6383739 : -16345834/6383739 : 1) C2b (83055358/45027577 : 380931/2648681 : 1)
** u= 369/148 ; tau(u)= 73/221 ; 38479*x^2 - 92353*y^2 + 141490*x*z + 38479*z^2
(-33769/150959 : 46476/150959 : 1) C1a (4269650485/39184623 : 191432095/39184623 : 1)
** u= 374/109 ; tau(u)= 156/265 ; -574*x^2 - 116114*y^2 + 164212*x*z - 574*z^2
(72131/469892 : -216371/469892 : 1) C1a (-256871/992854 : -44103/992854 : 1)
** u= 378/85 ; tau(u)= 208/293 ; -28814*x^2 - 128434*y^2 + 186148*x*z - 28814*z^2
(1171/3627 : 1702/3627 : 1) C1a (2009961834/1931906797 : 111496273/1931906797 : 1)
** u= 378/229 ; tau(u)= -80/149 ; 98482*x^2 - 38002*y^2 + 149284*x*z + 98482*z^2
(-1741/2027 : 2154/2027 : 1) C1a (-3130358/143361 : -217537/143361 : 1)
** u= 378/257 ; tau(u)= -136/121 ; 113602*x^2 - 10786*y^2 + 161380*x*z + 113602*z^2
(14557/32083 : 141636/32083 : 1) C1a (-23711139/2009441 : -3122533/2009441 : 1)
** u= 378/265 ; tau(u)= -152/113 ; 117346*x^2 - 2434*y^2 + 165988*x*z + 117346*z^2
(-335/643 : 3264/643 : 1) C1a (-21058/88191 : 22319/88191 : 1)
** u= 381/68 ; tau(u)= 245/313 ; -50777*x^2 - 135913*y^2 + 205186*x*z - 50777*z^2
(510127/335739 : -345296/335739 : 1) C1a (237025386/155381203 : 11062709/155381203 : 1)
** u= 381/200 ; tau(u)= -19/181 ; 79639*x^2 - 65161*y^2 + 145522*x*z + 79639*z^2
(-2985/1771 : -244/253 : 1) C1a (-162928977/11841077 : 291877/408313 : 1)
** u= 382/269 ; tau(u)= -156/113 ; 120386*x^2 - 1202*y^2 + 170260*x*z + 120386*z^2
(-235476/547187 : 4158125/547187 : 1) C1a (29482/170227 : 82653/170227 : 1)
** u= 386/97 ; tau(u)= 192/289 ; -18046*x^2 - 130178*y^2 + 185860*x*z - 18046*z^2
(451/3799 : -646/3799 : 1) C1a (-2413127/1219378 : -121203/1219378 : 1)
** u= 387/98 ; tau(u)= 191/289 ; -17273*x^2 - 130561*y^2 + 186250*x*z - 17273*z^2
(9601/87707 : 13090/87707 : 1) C1a (-15788294/4900641 : 731897/4900641 : 1)
** u= -388/369 ; tau(u)= 1126/757 ; -995554*x^2 + 121778*y^2 + 1418420*x*z - 995554*z^2
(4934/1247 : 11835/1247 : 1) C2b (7372451/2490326 : 727247/2490326 : 1)
** u= 389/164 ; tau(u)= 61/225 ; 50071*x^2 - 97529*y^2 + 155042*x*z + 50071*z^2
(953/961 : -1548/961 : 1) C1a (-37871066/17545277 : 1650653/17545277 : 1)
** u= 393/256 ; tau(u)= -119/137 ; 116911*x^2 - 23377*y^2 + 168610*x*z + 116911*z^2
(44767/1077983 : -2483872/1077983 : 1) C1a (-177745706/88798741 : 12869863/88798741 : 1)
** u= 397/244 ; tau(u)= -91/153 ; 110791*x^2 - 38537*y^2 + 165890*x*z + 110791*z^2
(-7123/24097 : 32808/24097 : 1) C1a (-37885/16473 : -2215/16473 : 1)
** u= 399/50 ; tau(u)= 299/349 ; -84401*x^2 - 154201*y^2 + 248602*x*z - 84401*z^2
(13/23 : 10/23 : 1) C1a (-18950029/26445353 : 1715783/26445353 : 1)
** u= 399/274 ; tau(u)= -149/125 ; 127951*x^2 - 9049*y^2 + 181402*x*z + 127951*z^2
(76391/15829 : -332110/15829 : 1) C1a (321811/78846 : -61723/78846 : 1)
407
>
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。
[参考文献]
- [1]Noam Elkies, "On A^4+B^4+C^4=D^4", Math Comp. 51(184), p824-835, 1988.
- [2]StarkExchange MATHEMATICS, "Distribution of Primitive Pytagorean Triples (PPT) and of solutions of A^4+B^4+C^4=D^4", 2016/07/08.
- [3]StarkExchange MATHEMATICS, "More elliptic curves for x^4+y^4+z^4=1?", 2017/07/28.
- [4]Tom Womack, "The quartic surfaces x^4+y^4+z^4=N", 2013/05/17.
- [5]Tom Womack, "elk18.mag", 2013/06/07.
- [6]Tom Womack, "elk18.pts", 2013/06/07.
- [7]Tom Womack, "Integer points on x^4+y^4+z^4=Nt^4", 2013/06/07.
- [8]StarkExchange MATHEMATICS, "a^4+b^4+c^4=2*d^2 such that a,b,c,d are all nonzero Integers & a+b+c!=0", 2024/04/26.
| Last Update: 2025.12.27 |
| H.Nakao |