Integer Points on A^4+B^4+C^4=338*D^4

[2026.02.01]A^4+B^4+C^4=338*D^4の整点

syzygyによる変換を計算するプログラム(Pari/GP)
A^4+B^4+C^4=2m^2D^4の整点を計算するプログラム(Pari/GP)
2次曲線(3a),(3b)がnon-singularであり、共に有理点を持つ有理数uを抽出するプログラム(MAGMA)
楕円曲線の4-descentから有理点を計算するプログラム(MAGMA)

■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。

338=2*13^2であるので、以下では、n=13とする。

■n=13のとき、２次曲線(**)は、有理点(1/3,1/3)を持つことが確認できる。

{MAGMAでの計算]

> P2 := ProjectiveSpace(Rationals(), 2);
> N:=13;
> C := Conic(P2,-N*y^2+x^2+x*z+z^2);
> HasRationalPoint(C);
true (1/3 : 1/3 : 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200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208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(13,1,200);
** u= 1/181 ; tau(u)= 361/180 ; -64799*x^2 + 65521*y^2 + 130322*x*z - 64799*z^2
  (47657/40315 : -5664/40315 : 1)  C1b (6472237/191247 : 1764433/191247 : 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 (692/313 : -219/313 : 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 (559/644 : -171/644 : 1)
** u= 3/101 ; tau(u)= 199/98 ; -19199*x^2 + 20393*y^2 + 39610*x*z - 19199*z^2
  (7797/10837 : 1918/10837 : 1)  C1b (509731/254689 : 121879/254689 : 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 (2419/296 : -781/296 : 1)
** u= -7/13 ; tau(u)= 33/20 ; -751*x^2 + 289*y^2 + 1138*x*z - 751*z^2
  (5/3 : 92/51 : 1)  C2b (-236/19 : 1647/323 : 1)
** u= 7/41 ; tau(u)= 75/34 ; -2263*x^2 + 3313*y^2 + 5674*x*z - 2263*z^2
  (7409/3381 : 1550/3381 : 1)  C1b (-63377/40933 : 22611/40933 : 1)
** u= -7/157 ; tau(u)= 321/164 ; -53743*x^2 + 49249*y^2 + 103090*x*z - 53743*z^2
  (10141/7671 : -3688/7671 : 1)  C2b (-35999/1404715 : -80799/280943 : 1)
** u= 8/17 ; tau(u)= 26/9 ; -98*x^2 + 514*y^2 + 740*x*z - 98*z^2
  (31/956 : -363/956 : 1)  C1b (49505/48597 : 15205/48597 : 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 (-19483/6305 : -1545/1261 : 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 (-2261/444 : 641/444 : 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 (-1909367/1483732 : 773931/1483732 : 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 (12060/7177 : 2945/7177 : 1)
** u= 9/13 ; tau(u)= 17/4 ; 49*x^2 + 257*y^2 + 370*x*z + 49*z^2
  (-167/103 : -132/103 : 1)  C1b (-68927/45028 : 18161/45028 : 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 (-321/59200 : -3473/11840 : 1)
** u= 9/169 ; tau(u)= 329/160 ; -51119*x^2 + 57041*y^2 + 108322*x*z - 51119*z^2
  (2815/1753 : -696/1753 : 1)  C1b (-354052/184053 : 126973/184053 : 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 (-32683/8348 : -232361/192004 : 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 (-960/931 : -265/931 : 1)
** u= -12/13 ; tau(u)= 38/25 ; -1106*x^2 + 194*y^2 + 1588*x*z - 1106*z^2
  (-32/11 : -97/11 : 1)  C2b (296/347 : 143/347 : 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 (1115/728 : 275/728 : 1)
** u= -12/109 ; tau(u)= 230/121 ; -29138*x^2 + 23618*y^2 + 53044*x*z - 29138*z^2
  (-243/1024 : -9713/7168 : 1)  C2b (12689/4448 : -22423/31136 : 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 (51536/49281 : 14023/49281 : 1)
** u= -12/185 ; tau(u)= 382/197 ; -77474*x^2 + 68306*y^2 + 146068*x*z - 77474*z^2
  (311/402 : -1121/2814 : 1)  C2b (9520168/217191 : 18884267/1520337 : 1)
** u= 13/8 ; tau(u)= -3/5 ; 119*x^2 - 41*y^2 + 178*x*z + 119*z^2
  (-3 : -4 : 1)  C1a (764/87 : 337/87 : 1)
** u= 13/49 ; tau(u)= 85/36 ; -2423*x^2 + 4633*y^2 + 7394*x*z - 2423*z^2
  (-337/601 : -756/601 : 1)  C1b (-401709/68812 : -106339/68812 : 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 (-379331/1999 : -94809/1999 : 1)
** u= 13/157 ; tau(u)= 301/144 ; -41303*x^2 + 49129*y^2 + 90770*x*z - 41303*z^2
  (173/5923 : -5256/5923 : 1)  C1b (-58980/25711 : 19805/25711 : 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 (-19440/5081 : -5725/5081 : 1)
** u= 15/73 ; tau(u)= 131/58 ; -6503*x^2 + 10433*y^2 + 17386*x*z - 6503*z^2
  (8947/20739 : -2962/20739 : 1)  C1b (107501/57312 : -25609/57312 : 1)
** u= 15/97 ; tau(u)= 179/82 ; -13223*x^2 + 18593*y^2 + 32266*x*z - 13223*z^2
  (22805/84909 : 46226/84909 : 1)  C1b (64128/176371 : -41617/176371 : 1)
** u= 16/113 ; tau(u)= 210/97 ; -18562*x^2 + 25282*y^2 + 44356*x*z - 18562*z^2
  (61/123 : -26/123 : 1)  C1b (-3259052/6511063 : -2185371/6511063 : 1)
** u= 17/4 ; tau(u)= 9/13 ; -49*x^2 - 257*y^2 + 370*x*z - 49*z^2
  (61/49 : -8/7 : 1)  C1a (49505/48597 : 15205/48597 : 1)
** u= -17/145 ; tau(u)= 307/162 ; -52199*x^2 + 41761*y^2 + 94538*x*z - 52199*z^2
  (1/17 : 18/17 : 1)  C2b (571072/119571 : 151903/119571 : 1)
** u= -21/29 ; tau(u)= 79/50 ; -4559*x^2 + 1241*y^2 + 6682*x*z - 4559*z^2
  (-940973/973083 : 3414890/973083 : 1)  C2b (-340959/71227 : 178853/71227 : 1)
** u= 21/37 ; tau(u)= 53/16 ; -71*x^2 + 2297*y^2 + 3250*x*z - 71*z^2
  (9/431 : 16/431 : 1)  C1b (172/325 : 17/65 : 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 (216812/74637 : 50827/74637 : 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 (1430848/45813 : 432833/45813 : 1)
** u= 24/25 ; tau(u)= 26 ; 574*x^2 + 674*y^2 + 1252*x*z + 574*z^2
  (-109/166 : 5/166 : 1)  C1b (-2308/10379 : 2551/10379 : 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 (-1716/907 : 839/907 : 1)
** u= -24/73 ; tau(u)= 170/97 ; -18242*x^2 + 10082*y^2 + 29476*x*z - 18242*z^2
  (-1/3 : -368/213 : 1)  C2b (-644/363 : -21767/25773 : 1)
** u= -25/121 ; tau(u)= 267/146 ; -42007*x^2 + 28657*y^2 + 71914*x*z - 42007*z^2
  (-8087/164833 : 208010/164833 : 1)  C2b (-65225288/53713453 : 32926197/53713453 : 1)
** u= -25/137 ; tau(u)= 299/162 ; -51863*x^2 + 36913*y^2 + 90026*x*z - 51863*z^2
  (-3851/154349 : 6030/4979 : 1)  C2b (-2053487/10551611 : 3630563/10551611 : 1)
** u= 25/169 ; tau(u)= 313/144 ; -40847*x^2 + 56497*y^2 + 98594*x*z - 40847*z^2
  (-52699/184291 : -1460160/1290037 : 1)  C1b (-203284/37077 : -401213/259539 : 1)
** u= 26 ; tau(u)= 24/25 ; -574*x^2 - 674*y^2 + 1252*x*z - 574*z^2
  (109/166 : -5/166 : 1)  C1a (1244/259 : 307/259 : 1)
** u= 26/9 ; tau(u)= 8/17 ; 98*x^2 - 514*y^2 + 740*x*z + 98*z^2
  (-113/1861 : -600/1861 : 1)  C1a (-68927/45028 : 18161/45028 : 1)
** u= 27/157 ; tau(u)= 287/130 ; -33071*x^2 + 48569*y^2 + 83098*x*z - 33071*z^2
  (25509/57425 : -13498/57425 : 1)  C1b (23174256/1575083 : -5795831/1575083 : 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 (4672/2329 : -153/137 : 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 (-51/568 : 187/568 : 1)
** u= 32/185 ; tau(u)= 338/153 ; -45794*x^2 + 67426*y^2 + 115268*x*z - 45794*z^2
  (9421/177077 : 136032/177077 : 1)  C1b (-118907/29769 : 34247/29769 : 1)
** u= 33/20 ; tau(u)= -7/13 ; 751*x^2 - 289*y^2 + 1138*x*z + 751*z^2
  (-1/583 : 15956/9911 : 1)  C1a (6479/643 : -45783/10931 : 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 (145999/1013 : -34173/1013 : 1)
** u= -33/89 ; tau(u)= 211/122 ; -28679*x^2 + 14753*y^2 + 45610*x*z - 28679*z^2
  (59971/478031 : -602146/478031 : 1)  C2b (-22085/32712 : 17125/32712 : 1)
** u= 33/181 ; tau(u)= 329/148 ; -42719*x^2 + 64433*y^2 + 109330*x*z - 42719*z^2
  (5769/5081071 : 4131236/5081071 : 1)  C1b (3761156/1181921 : -887503/1181921 : 1)
** u= -35/37 ; tau(u)= 109/72 ; -9143*x^2 + 1513*y^2 + 13106*x*z - 9143*z^2
  (-253/2209 : -5892/2209 : 1)  C2b (-261972/9371 : -155033/9371 : 1)
** u= -36/85 ; tau(u)= 206/121 ; -27986*x^2 + 13154*y^2 + 43732*x*z - 27986*z^2
  (2/9 : 11/9 : 1)  C2b (202384/88603 : 57503/88603 : 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 (-15853/3159 : -4357/3159 : 1)
** u= 38/25 ; tau(u)= -12/13 ; 1106*x^2 - 194*y^2 + 1588*x*z + 1106*z^2
  (-3/2 : 5/2 : 1)  C1a (-37/57 : 23/57 : 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 (1728/2011 : 721/2011 : 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 (-47423/13147 : -12133/13147 : 1)
** u= 40/53 ; tau(u)= 66/13 ; 1262*x^2 + 4018*y^2 + 5956*x*z + 1262*z^2
  (-3 : -8/7 : 1)  C1b (8884/2827 : 16509/19789 : 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 (668/65751 : 22277/65751 : 1)
** u= 40/121 ; tau(u)= 202/81 ; -11522*x^2 + 27682*y^2 + 42404*x*z - 11522*z^2
  (16153/2909 : 6336/2909 : 1)  C1b (-7271324/4760987 : 2357731/4760987 : 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 (-150676/83119 : -76523/83119 : 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 (3344287/1151531 : -784403/1151531 : 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 (-18836/43431 : -10241/43431 : 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 (-645569/330264 : -416641/330264 : 1)
** u= 45/109 ; tau(u)= 173/64 ; -6167*x^2 + 21737*y^2 + 31954*x*z - 6167*z^2
  (-58819/76259 : -96048/76259 : 1)  C1b (234367/78429 : -55247/78429 : 1)
** u= 48/193 ; tau(u)= 338/145 ; -39746*x^2 + 72194*y^2 + 116548*x*z - 39746*z^2
  (15393/1559 : -9646/1559 : 1)  C1b (1531409/82813 : 375063/82813 : 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 (-455933/170761 : -115203/170761 : 1)
** u= 49/193 ; tau(u)= 337/144 ; -39071*x^2 + 72097*y^2 + 115970*x*z - 39071*z^2
  (42299/120541 : -25344/120541 : 1)  C1b (-52301573/38193833 : 18513623/38193833 : 1)
** u= 51/125 ; tau(u)= 199/74 ; -8351*x^2 + 28649*y^2 + 42202*x*z - 8351*z^2
  (-107/1017 : -682/1017 : 1)  C1b (-941747/40207 : 226249/40207 : 1)
** u= -51/149 ; tau(u)= 349/200 ; -77399*x^2 + 41801*y^2 + 124402*x*z - 77399*z^2
  (-515019/2097679 : -3442940/2097679 : 1)  C2b (20453/4431 : 6101/4431 : 1)
** u= 51/157 ; tau(u)= 263/106 ; -19871*x^2 + 46697*y^2 + 71770*x*z - 19871*z^2
  (-19377/197047 : 1051174/1379329 : 1)  C1b (39552/9643 : 64721/67501 : 1)
** u= -52/121 ; tau(u)= 294/173 ; -57154*x^2 + 26578*y^2 + 89140*x*z - 57154*z^2
  (1642/157 : -2233/157 : 1)  C2b (-177464143/6594917 : -65282109/6594917 : 1)
** u= -52/193 ; tau(u)= 438/245 ; -117346*x^2 + 71794*y^2 + 194548*x*z - 117346*z^2
  (3519/5042 : -3703/5042 : 1)  C2b (2610728/3522263 : -920457/3522263 : 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 (-2261/444 : 641/444 : 1)
** u= 53/16 ; tau(u)= 21/37 ; 71*x^2 - 2297*y^2 + 3250*x*z + 71*z^2
  (-43/4101 : -520/4101 : 1)  C1a (8429/8996 : -2907/8996 : 1)
** u= 55/73 ; tau(u)= 91/18 ; 2377*x^2 + 7633*y^2 + 11306*x*z + 2377*z^2
  (-5/11 : 6/11 : 1)  C1b (-316479/54992 : -73901/54992 : 1)
** u= -56/61 ; tau(u)= 178/117 ; -24242*x^2 + 4306*y^2 + 34820*x*z - 24242*z^2
  (3613/4697 : 7776/4697 : 1)  C2b (40351/7596 : -19789/7596 : 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 (-341156/307417 : 102609/307417 : 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 (32204/44201 : -12331/44201 : 1)
** u= 57/125 ; tau(u)= 193/68 ; -5999*x^2 + 28001*y^2 + 40498*x*z - 5999*z^2
  (5327/49201 : 12068/49201 : 1)  C1b (605031/712924 : -203783/712924 : 1)
** u= 57/185 ; tau(u)= 313/128 ; -29519*x^2 + 65201*y^2 + 101218*x*z - 29519*z^2
  (2864959/32625105 : -18452848/32625105 : 1)  C1b (8782204/5414253 : -2167813/5414253 : 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 (78344/27383 : 18369/27383 : 1)
** u= 60/89 ; tau(u)= 118/29 ; 1918*x^2 + 12242*y^2 + 17524*x*z + 1918*z^2
  (-586/4903 : 541/4903 : 1)  C1b (-2353/1843 : -663/1843 : 1)
** u= 60/109 ; tau(u)= 158/49 ; -1202*x^2 + 20162*y^2 + 28564*x*z - 1202*z^2
  (2657/63094 : 497/63094 : 1)  C1b (-39189/87127 : -22649/87127 : 1)
** u= 60/121 ; tau(u)= 182/61 ; -3842*x^2 + 25682*y^2 + 36724*x*z - 3842*z^2
  (-934/793 : 1133/793 : 1)  C1b (156392/4179 : -36587/4179 : 1)
** u= -60/121 ; tau(u)= 302/181 ; -61922*x^2 + 25682*y^2 + 94804*x*z - 61922*z^2
  (-2016/10337 : -18557/10337 : 1)  C2b (429784/808013 : 233481/808013 : 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 (103075768/15369363 : 32103859/15369363 : 1)
** u= -63/65 ; tau(u)= 193/128 ; -28799*x^2 + 4481*y^2 + 41218*x*z - 28799*z^2
  (9/7 : 16/7 : 1)  C2b (11509/9036 : -4933/9036 : 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 (12799/936 : 2993/936 : 1)
** u= -64/53 ; tau(u)= 170/117 ; -23282*x^2 + 1522*y^2 + 32996*x*z - 23282*z^2
  (299/487 : -1356/487 : 1)  C2b (431436/73781 : 349697/73781 : 1)
** u= -65/97 ; tau(u)= 259/162 ; -48263*x^2 + 14593*y^2 + 71306*x*z - 48263*z^2
  (-42871/387625 : -764334/387625 : 1)  C2b (-1047536/227111 : -525781/227111 : 1)
** u= 66/13 ; tau(u)= 40/53 ; -1262*x^2 - 4018*y^2 + 5956*x*z - 1262*z^2
  (30/133 : 61/931 : 1)  C1a (-3244/463 : -5637/3241 : 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 (4301/871 : -1581/871 : 1)
** u= -69/125 ; tau(u)= 319/194 ; -70511*x^2 + 26489*y^2 + 106522*x*z - 70511*z^2
  (5657/7837 : -8390/7837 : 1)  C2b (448949/131768 : 147467/131768 : 1)
** u= -69/149 ; tau(u)= 367/218 ; -90287*x^2 + 39641*y^2 + 139450*x*z - 90287*z^2
  (833/8017 : -78098/56119 : 1)  C2b (-24784/60711 : -203533/424977 : 1)
** u= 69/197 ; tau(u)= 325/128 ; -28007*x^2 + 72857*y^2 + 110386*x*z - 28007*z^2
  (10011/313301 : -181712/313301 : 1)  C1b (-123422444/3252347 : 30002811/3252347 : 1)
** u= 72/85 ; tau(u)= 98/13 ; 4846*x^2 + 9266*y^2 + 14788*x*z + 4846*z^2
  (-599/253 : 144/253 : 1)  C1b (-30116/20971 : 7567/20971 : 1)
** u= -72/85 ; tau(u)= 242/157 ; -44114*x^2 + 9266*y^2 + 63748*x*z - 44114*z^2
  (3316/589 : -6369/589 : 1)  C2b (86628/67829 : 32423/67829 : 1)
** u= -72/149 ; tau(u)= 370/221 ; -92498*x^2 + 39218*y^2 + 142084*x*z - 92498*z^2
  (1673/26769 : -39172/26769 : 1)  C2b (-53236/60837 : -38707/60837 : 1)
** u= 72/181 ; tau(u)= 290/109 ; -18578*x^2 + 60338*y^2 + 89284*x*z - 18578*z^2
  (3383/29435 : 1584/4205 : 1)  C1b (63681/17797 : -14899/17797 : 1)
** u= 75/34 ; tau(u)= 7/41 ; 2263*x^2 - 3313*y^2 + 5674*x*z + 2263*z^2
  (241/51 : 250/51 : 1)  C1a (36809/7576 : -10377/7576 : 1)
** u= 75/149 ; tau(u)= 223/74 ; -5327*x^2 + 38777*y^2 + 55354*x*z - 5327*z^2
  (3217/40401 : -6334/40401 : 1)  C1b (54136/19903 : -13087/19903 : 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 (166319/164637 : 50023/164637 : 1)
** u= -76/73 ; tau(u)= 222/149 ; -38626*x^2 + 4882*y^2 + 55060*x*z - 38626*z^2
  (436/231 : 127/33 : 1)  C2b (3079/3064 : -1563/3064 : 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 (1044817/709552 : -269499/709552 : 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 (137672/4987 : 106329/4987 : 1)
** u= 79/50 ; tau(u)= -21/29 ; 4559*x^2 - 1241*y^2 + 6682*x*z + 4559*z^2
  (-3343/5367 : -7090/5367 : 1)  C1a (1432/1209 : -1103/1209 : 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 (171131/177873 : -68323/177873 : 1)
** u= 81/101 ; tau(u)= 121/20 ; 5761*x^2 + 13841*y^2 + 21202*x*z + 5761*z^2
  (-20135/15163 : -14256/15163 : 1)  C1b (106876/258913 : -74219/258913 : 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 (12944/8279 : 20419/8279 : 1)
** u= 84/85 ; tau(u)= 86 ; 7054*x^2 + 7394*y^2 + 14452*x*z + 7054*z^2
  (-1842/2107 : -337/2107 : 1)  C1b (-3164008/897643 : -777639/897643 : 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 (66999/25211 : 32063/25211 : 1)
** u= 84/157 ; tau(u)= 230/73 ; -3602*x^2 + 42242*y^2 + 59956*x*z - 3602*z^2
  (-83/164 : -149/164 : 1)  C1b (707624728/125701901 : -166175337/125701901 : 1)
** u= 84/173 ; tau(u)= 262/89 ; -8786*x^2 + 52802*y^2 + 75700*x*z - 8786*z^2
  (-872/503 : -893/503 : 1)  C1b (-1342632/432475 : 68453/86495 : 1)
** u= 84/193 ; tau(u)= 302/109 ; -16706*x^2 + 67442*y^2 + 98260*x*z - 16706*z^2
  (-873/13502 : -7907/13502 : 1)  C1b (798349/1520 : 37785/304 : 1)
** u= 85/36 ; tau(u)= 13/49 ; 2423*x^2 - 4633*y^2 + 7394*x*z + 2423*z^2
  (-1163/3193 : -336/3193 : 1)  C1a (-30116/20971 : 7567/20971 : 1)
** u= 86 ; tau(u)= 84/85 ; -7054*x^2 - 7394*y^2 + 14452*x*z - 7054*z^2
  (4070/5073 : 29/5073 : 1)  C1a (-1848263/561619 : -597237/561619 : 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 (2488/1287 : -791/1287 : 1)
** u= 87/89 ; tau(u)= 91/2 ; 7561*x^2 + 8273*y^2 + 15850*x*z + 7561*z^2
  (-5061/4271 : -1154/4271 : 1)  C1b (72128/3793 : 20091/3793 : 1)
** u= -87/173 ; tau(u)= 433/260 ; -127631*x^2 + 52289*y^2 + 195058*x*z - 127631*z^2
  (-119159/68487 : 276676/68487 : 1)  C2b (132348743/19779404 : 45582623/19779404 : 1)
** u= 88/149 ; tau(u)= 210/61 ; 302*x^2 + 36658*y^2 + 51844*x*z + 302*z^2
  (-10815/177302 : -49517/177302 : 1)  C1b (-214748/259129 : 78267/259129 : 1)
** u= 91/2 ; tau(u)= 87/89 ; -7561*x^2 - 8273*y^2 + 15850*x*z - 7561*z^2
  (2283/2837 : 538/2837 : 1)  C1a (-2485737/28400 : -27139/1136 : 1)
** u= 91/18 ; tau(u)= 55/73 ; -2377*x^2 - 7633*y^2 + 11306*x*z - 2377*z^2
  (12541/9271 : 9822/9271 : 1)  C1a (-5417097/1876411 : -1454591/1876411 : 1)
** u= -92/113 ; tau(u)= 318/205 ; -75586*x^2 + 17074*y^2 + 109588*x*z - 75586*z^2
  (125315/53844 : 28229/7692 : 1)  C2b (-3168467/1746256 : -2261691/1746256 : 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 (-40079983/3104861 : -9375393/3104861 : 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 (-44764/44809 : 12801/44809 : 1)
** u= 93/97 ; tau(u)= 101/4 ; 8617*x^2 + 10169*y^2 + 18850*x*z + 8617*z^2
  (-3733/4237 : -1516/4237 : 1)  C1b (11149/3900 : -11/12 : 1)
** u= 96/113 ; tau(u)= 130/17 ; 8638*x^2 + 16322*y^2 + 26116*x*z + 8638*z^2
  (-3161/1527 : 1096/1527 : 1)  C1b (154516/21743 : 40437/21743 : 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 (-2356/1279 : -70527/152201 : 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 (1370716/327187 : 336789/327187 : 1)
** u= 97/169 ; tau(u)= 241/72 ; -959*x^2 + 47713*y^2 + 67490*x*z - 959*z^2
  (10487/1103293 : -90012/1103293 : 1)  C1b (-4007148/665047 : -949609/665047 : 1)
** u= 98/13 ; tau(u)= 72/85 ; -4846*x^2 - 9266*y^2 + 14788*x*z - 4846*z^2
  (57/22 : 7/22 : 1)  C1a (-401709/68812 : -106339/68812 : 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 (-18836/43431 : -10241/43431 : 1)
** u= 99/101 ; tau(u)= 103/2 ; 9793*x^2 + 10601*y^2 + 20410*x*z + 9793*z^2
  (-1572997/1988591 : 289110/1988591 : 1)  C1b (111039/168181 : 65837/168181 : 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 (1324348/442179 : -566299/442179 : 1)
** u= 99/125 ; tau(u)= 151/26 ; 8449*x^2 + 21449*y^2 + 32602*x*z + 8449*z^2
  (-3583/10137 : 3110/10137 : 1)  C1b (94992/104659 : -38329/104659 : 1)
** u= 101/4 ; tau(u)= 93/97 ; -8617*x^2 - 10169*y^2 + 18850*x*z - 8617*z^2
  (28253/39477 : 8396/39477 : 1)  C1a (-61596/8125 : 3509/1625 : 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 (36724/1891 : 9009/1891 : 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 (380116/45487 : 97683/45487 : 1)
** u= 103/2 ; tau(u)= 99/101 ; -9793*x^2 - 10601*y^2 + 20410*x*z - 9793*z^2
  (524483/660417 : -99266/660417 : 1)  C1a (22208/100087 : -24907/100087 : 1)
** u= -105/137 ; tau(u)= 379/242 ; -106103*x^2 + 26513*y^2 + 154666*x*z - 106103*z^2
  (15/7 : 22/7 : 1)  C2b (1033288/619621 : 358613/619621 : 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 (412207/40036 : -98059/40036 : 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 (2616/73625 : -1309/2945 : 1)
** u= 108/173 ; tau(u)= 238/65 ; 3214*x^2 + 48194*y^2 + 68308*x*z + 3214*z^2
  (-1276/24773 : -1941/24773 : 1)  C1b (-1056409/1150608 : 356051/1150608 : 1)
** u= 108/193 ; tau(u)= 278/85 ; -2786*x^2 + 62834*y^2 + 88948*x*z - 2786*z^2
  (32/21655 : -4451/21655 : 1)  C1b (312224/21717 : -72871/21717 : 1)
** u= 109/72 ; tau(u)= -35/37 ; 9143*x^2 - 1513*y^2 + 13106*x*z + 9143*z^2
  (253/2209 : -5892/2209 : 1)  C1a (-17548/16397 : -7643/16397 : 1)
** u= 109/145 ; tau(u)= 181/36 ; 9289*x^2 + 30169*y^2 + 44642*x*z + 9289*z^2
  (-26597/95081 : 3888/13583 : 1)  C1b (-435508/157599 : 102871/157599 : 1)
** u= -111/97 ; tau(u)= 305/208 ; -74207*x^2 + 6497*y^2 + 105346*x*z - 74207*z^2
  (5481/14951 : -39592/14951 : 1)  C2b (14447/91764 : -64937/91764 : 1)
** u= 112/121 ; tau(u)= 130/9 ; 12382*x^2 + 16738*y^2 + 29444*x*z + 12382*z^2
  (-347/203 : 66/203 : 1)  C1b (3146452/770241 : 920801/770241 : 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 (73804/76063 : -28287/76063 : 1)
** u= -113/193 ; tau(u)= 499/306 ; -174503*x^2 + 61729*y^2 + 261770*x*z - 174503*z^2
  (-876041/2476379 : -5357718/2476379 : 1)  C2b (-45352672/6568803 : 20181617/6568803 : 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 (388/18773 : 4389/18773 : 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 (-925472/68067 : 231121/68067 : 1)
** u= -115/157 ; tau(u)= 429/272 ; -134743*x^2 + 36073*y^2 + 197266*x*z - 134743*z^2
  (5097/191 : 9584/191 : 1)  C2b (11078284/2612221 : 4351251/2612221 : 1)
** u= 117/125 ; tau(u)= 133/8 ; 13561*x^2 + 17561*y^2 + 31378*x*z + 13561*z^2
  (-6993/5477 : 2740/5477 : 1)  C1b (349684/48343 : -3373/1667 : 1)
** u= 118/29 ; tau(u)= 60/89 ; -1918*x^2 - 12242*y^2 + 17524*x*z - 1918*z^2
  (1780/14029 : -2101/14029 : 1)  C1a (-132219/24232 : 32147/24232 : 1)
** u= -120/101 ; tau(u)= 322/221 ; -83282*x^2 + 6002*y^2 + 118084*x*z - 83282*z^2
  (821/4465 : 14624/4465 : 1)  C2b (691/863 : -537/863 : 1)
** u= 121/20 ; tau(u)= 81/101 ; -5761*x^2 - 13841*y^2 + 21202*x*z - 5761*z^2
  (1057/2729 : 924/2729 : 1)  C1a (-7271324/4760987 : 2357731/4760987 : 1)
** u= -127/197 ; tau(u)= 521/324 ; -193823*x^2 + 61489*y^2 + 287570*x*z - 193823*z^2
  (19667/13021 : 23580/13021 : 1)  C2b (38783/223485 : 16915/44697 : 1)
** u= 128/173 ; tau(u)= 218/45 ; 12334*x^2 + 43474*y^2 + 63908*x*z + 12334*z^2
  (-379/1447 : -414/1447 : 1)  C1b (-14033/4667 : 3307/4667 : 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 (-34724/6047 : 8169/6047 : 1)
** u= 130/9 ; tau(u)= 112/121 ; -12382*x^2 - 16738*y^2 + 29444*x*z - 12382*z^2
  (754/1237 : -297/1237 : 1)  C1a (9481/25444 : 6013/25444 : 1)
** u= 130/17 ; tau(u)= 96/113 ; -8638*x^2 - 16322*y^2 + 26116*x*z - 8638*z^2
  (59271/31369 : 24392/31369 : 1)  C1a (14321/1579 : -3443/1579 : 1)
** u= 131/58 ; tau(u)= 15/73 ; 6503*x^2 - 10433*y^2 + 17386*x*z + 6503*z^2
  (17971/36117 : -45782/36117 : 1)  C1a (26688/17869 : -9433/17869 : 1)
** u= -132/137 ; tau(u)= 406/269 ; -127298*x^2 + 20114*y^2 + 182260*x*z - 127298*z^2
  (6792/5753 : -12139/5753 : 1)  C2b (4781751/2687224 : 2056663/2687224 : 1)
** u= 132/197 ; tau(u)= 262/65 ; 8974*x^2 + 60194*y^2 + 86068*x*z + 8974*z^2
  (-213/100 : 149/100 : 1)  C1b (-327512/291893 : 97407/291893 : 1)
** u= 133/8 ; tau(u)= 117/125 ; -13561*x^2 - 17561*y^2 + 31378*x*z - 13561*z^2
  (87913/147317 : 20340/147317 : 1)  C1a (-40797/31511 : -16093/31511 : 1)
** u= -133/101 ; tau(u)= 335/234 ; -91823*x^2 + 2713*y^2 + 129914*x*z - 91823*z^2
  (15973/2245673 : 12999066/2245673 : 1)  C2b (28863/34381 : -33583/34381 : 1)
** u= -135/109 ; tau(u)= 353/244 ; -100847*x^2 + 5537*y^2 + 142834*x*z - 100847*z^2
  (17/43 : -992/301 : 1)  C2b (363972/684259 : -3479311/4789813 : 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 (-1569943/80541 : -368731/80541 : 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 (64768/12601 : -25593/12601 : 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 (-19697/309452 : 77853/309452 : 1)
** u= -141/101 ; tau(u)= 343/242 ; -97247*x^2 + 521*y^2 + 137530*x*z - 97247*z^2
  (-201/2819 : 40502/2819 : 1)  C2b (-6182040/4071473 : -30293735/4071473 : 1)
** u= -141/197 ; tau(u)= 535/338 ; -208607*x^2 + 57737*y^2 + 306106*x*z - 208607*z^2
  (89541/315253 : 488254/315253 : 1)  C2b (527454936/92175077 : 212015221/92175077 : 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 (-20337/4381 : 6263/4381 : 1)
** u= -147/145 ; tau(u)= 437/292 ; -148919*x^2 + 20441*y^2 + 212578*x*z - 148919*z^2
  (275/2137 : 5264/2137 : 1)  C2b (159204492/18008371 : 592769/114703 : 1)
** u= -148/109 ; tau(u)= 366/257 ; -110194*x^2 + 1858*y^2 + 155860*x*z - 110194*z^2
  (-5784/2489 : -59659/2489 : 1)  C2b (17048/39653 : -54099/39653 : 1)
** u= 151/26 ; tau(u)= 99/125 ; -8449*x^2 - 21449*y^2 + 32602*x*z - 8449*z^2
  (269/599 : -274/599 : 1)  C1a (-848376/583909 : -277523/583909 : 1)
** u= 153/169 ; tau(u)= 185/16 ; 22897*x^2 + 33713*y^2 + 57634*x*z + 22897*z^2
  (-1353/2729 : 104/2729 : 1)  C1b (3865061/524583 : 1051793/524583 : 1)
** u= 153/185 ; tau(u)= 217/32 ; 21361*x^2 + 45041*y^2 + 70498*x*z + 21361*z^2
  (-33883/46451 : 29928/46451 : 1)  C1b (-1505193/20573 : -367811/20573 : 1)
** u= 158/49 ; tau(u)= 60/109 ; 1202*x^2 - 20162*y^2 + 28564*x*z + 1202*z^2
  (-101/13500 : 2989/13500 : 1)  C1a (-10599/8552 : 3113/8552 : 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 (-44201/32204 : 12331/32204 : 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 (220349/372616 : -157269/372616 : 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 (14648/17439 : 13577/17439 : 1)
** u= 168/181 ; tau(u)= 194/13 ; 27886*x^2 + 37298*y^2 + 65860*x*z + 27886*z^2
  (-35889/45491 : 19304/45491 : 1)  C1b (6118604/1991855 : 374855/398371 : 1)
** u= 170/97 ; tau(u)= -24/73 ; 18242*x^2 - 10082*y^2 + 29476*x*z + 18242*z^2
  (-373/876 : -58759/62196 : 1)  C1a (-28/183 : 3979/12993 : 1)
** u= 170/117 ; tau(u)= -64/53 ; 23282*x^2 - 1522*y^2 + 32996*x*z + 23282*z^2
  (3455/3413 : 24828/3413 : 1)  C1a (-795513/321268 : 559471/321268 : 1)
** u= 173/64 ; tau(u)= 45/109 ; 6167*x^2 - 21737*y^2 + 31954*x*z + 6167*z^2
  (-367/2049 : 352/2049 : 1)  C1a (-14033/4667 : 3307/4667 : 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 (188/443 : -12783/52717 : 1)
** u= 178/117 ; tau(u)= -56/61 ; 24242*x^2 - 4306*y^2 + 34820*x*z + 24242*z^2
  (3491/11033 : -1920/649 : 1)  C1a (-7596/40351 : -19789/40351 : 1)
** u= 179/82 ; tau(u)= 15/97 ; 13223*x^2 - 18593*y^2 + 32266*x*z + 13223*z^2
  (-59989/124253 : 24574/124253 : 1)  C1a (-8176757/4264728 : -1941347/4264728 : 1)
** u= 181/36 ; tau(u)= 109/145 ; -9289*x^2 - 30169*y^2 + 44642*x*z - 9289*z^2
  (49861/42493 : -42588/42493 : 1)  C1a (63681/17797 : -14899/17797 : 1)
** u= 182/61 ; tau(u)= 60/121 ; 3842*x^2 - 25682*y^2 + 36724*x*z + 3842*z^2
  (793/934 : -1133/934 : 1)  C1a (-904/24763 : -5789/24763 : 1)
** u= -183/157 ; tau(u)= 497/340 ; -197711*x^2 + 15809*y^2 + 280498*x*z - 197711*z^2
  (207/121 : 524/121 : 1)  C2b (89404/56919 : 52631/56919 : 1)
** u= 185/16 ; tau(u)= 153/169 ; -22897*x^2 - 33713*y^2 + 57634*x*z - 22897*z^2
  (17141/32909 : -5408/32909 : 1)  C1a (-118907/29769 : 34247/29769 : 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 (145684/7619 : -39807/7619 : 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 (-2775/4664 : -1145/4664 : 1)
** u= -189/173 ; tau(u)= 535/362 ; -226367*x^2 + 24137*y^2 + 321946*x*z - 226367*z^2
  (488305/7936293 : 23264642/7936293 : 1)  C2b (5814904/4247433 : 174337/249849 : 1)
** u= -191/169 ; tau(u)= 529/360 ; -222719*x^2 + 20641*y^2 + 316322*x*z - 222719*z^2
  (-132601/25986349 : -85670676/25986349 : 1)  C2b (-264389/188676 : -321851/188676 : 1)
** u= 193/68 ; tau(u)= 57/125 ; 5999*x^2 - 28001*y^2 + 40498*x*z + 5999*z^2
  (-3219/29321 : 7064/29321 : 1)  C1a (-1057324/248491 : 247573/248491 : 1)
** u= 193/128 ; tau(u)= -63/65 ; 28799*x^2 - 4481*y^2 + 41218*x*z + 28799*z^2
  (3877/12917 : -40368/12917 : 1)  C1a (-80444772/17153221 : -41368777/17153221 : 1)
** u= 194/13 ; tau(u)= 168/181 ; -27886*x^2 - 37298*y^2 + 65860*x*z - 27886*z^2
  (752/927 : -407/927 : 1)  C1a (1258876/591905 : 59521/118381 : 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 (258508/685177 : 5583431/11648009 : 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 (-35922697/712331 : -37419033/712331 : 1)
** u= 199/74 ; tau(u)= 51/125 ; 8351*x^2 - 28649*y^2 + 42202*x*z + 8351*z^2
  (107/1017 : 682/1017 : 1)  C1a (-1412872/351341 : 330093/351341 : 1)
** u= 199/98 ; tau(u)= 3/101 ; 19199*x^2 - 20393*y^2 + 39610*x*z + 19199*z^2
  (8521/6099 : -14294/6099 : 1)  C1a (-4761040/1757097 : 1148125/1757097 : 1)
208
>

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=-3/5のとき

1^4+3^4+4^4=338*1^4
1237^4+1943^4+2228^4=338*591^4
27221^4+36079^4+40804^4=338*11037^4
12897^4+25549^4+61708^4=338*14503^4
640609^4+3875723^4+5831828^4=338*1422123^4
37169179^4+39007473^4+46169012^4=338*12690827^4
19055497^4+136834381^4+258298844^4=338*61394181^4
57091397667^4+120558423329^4+147525150524^4=338*37874724929^4
2644647787489^4+11471523886909^4+23107596475676^4=338*5469470215953^4
3957641182169219^4+4327629783731433^4+5045941251037228^4=338*1385185958658067^4
16597034591065171^4+20053503476781713^4+22841679669114916^4=338*6231944505859329^4
3441660162794047^4+87717348183209333^4+145107041863726732^4=338*34919488113118779^4
228603292147551063^4+267354276949630411^4+790242495326356052^4=338*185221899351872063^4
938865586829981541^4+1825012170391742879^4+2187837241901099044^4=338*566382797928910957^4
74402036288882707173^4+134877640639777407871^4+159061946414005060796^4=338*41491306124231416829^4
1140966703427886431333^4+1698373384005296217319^4+1933282382355787608172^4=338*516122718109559089167^4
10222319979046099367477^4+11686528192285818476343^4+13455373951221713136308^4=338*3685362256226942354351^4
16319770494007895796570753^4+24850259950777162719155893^4+65772810207327062713143052^4=338*15431563703929104993714619^4
36561027699467462508679107^4+84254204687357859884117849^4+105403669498632599836197116^4=338*26847763472332280768857349^4
1262174324426791700720303743^4+12258791488868890606950797549^4+19329308600709610722379826188^4=338*4680258448977144638428156743^4
4557574596199691831690826319^4+85877587921520915090989939603^4+152200133404923449489172517444^4=338*36363702184096266836479322481^4
7107397654686307982075822804521^4+24758734538111669737283490881523^4+34085618119541225440387683691396^4=338*8456038272557114582115494792821^4
6113532410325295155175389475457993^4+27894045152684885125580263826978947^4+40329244502806022745077857386331852^4=338*9904053147234777005079599580789219^4
236165441833432584157097542150079759^4+559833388020279596175337986199786197^4+705368780097118843222237472218539076^4=338*179243293066474170381728327124294541^4
2673479184326283856599273935145572911^4+11401373799214602837893045554846282989^4+16300412471302520773400047099441359836^4=338*4011726630732584810317916427878429633^4
8405084816281454384789339795479359427^4+14941688974272402326743778577674087921^4+17544164385851440836087911913302989612^4=338*4586525636791840055010705012752822727^4
151324345678458428514240262730875979947^4+252768594433071342432995423210446894721^4+292992294358011389463835017405660130348^4=338*77151999731600629683346335745911427947^4
20837084530372574840708599973037248301131^4+26307548513037778449077351223905176405753^4+29817145131403175319380161429543135822844^4=338*8104106387080383968718932381476369906569^4
1004323441414360858207366099695139971049^4+52580469150315481529407886378577319418171^4+88306592550047422315441181451624666601436^4=338*21213842599929873800554625289222904264233^4
127440929297543323657386701851618648962779^4+179713296834017431534681863097695299758513^4+203585268399159216408160875543519838554932^4=338*54694282178043242561525208661911616312587^4
56146571216393796588354971182593297212711794301^4+66973959085958534650621312072997708996703906807^4+76440976249509107485276064851968421488856164212^4=338*20876425874763467303754199386709467042548698307^4
85406191924704390447148880060391407631454072543^4+249413349564622808240502212377453816697381009659^4+542404408890134359458650403621057052548732979396^4=338*127910630281036918984458731488852429563788931741^4
34030337752130712533287509166407911979790316056831^4+89525259028781903115735602488217965927780081249989^4+115745426748247402016021519117675121770997973826204^4=338*29180167289799414487590980361454872757304310128813^4
13141505016775336297885825146386281243731671815125945611^4+14301622165769034156870300399566753930546434302536560503^4+43713371273238199769656011669268133791333428069541028428^4=338*10244606864126525977442767879898875627537982202322321903^4
197810715588277166559493025139106020959211914573647253917^4+198737830580069928216148094426129526762484752921661293423^4+239860255779814443135129954986638238773600228888066315372^4=338*65968216459608694049945999733093794456787117819754243591^4
437464010835322688427177842768386611542774050591868058767774007^4+1682073678666617436268149455647191260973878441860427563096054787^4+3458639095711286651058562366134489423704382394004510270809053772^4=338*817735176368229926317582184234689866657511030722823797199706659^4
31499565971956184957398290068228858019618767911528938505695904307^4+46172225271511445002807835991664635164653745716080236500131638569^4+52471466843325399934529602540667908735973892989519862381596412476^4=338*14033625870678033910912419258097062279878506043655455494661370069^4
92498662896744768662880662481358129047345185184785849952202783647^4+95899169758795842117731858013448788194590156819684746983763593461^4+114082634654600508455543319116066046481141669858805266318089934828^4=338*31366360858424289564130391948689415366933797111842452475397792747^4
11836040178323053845124833162992618674715037293222582018125152584511^4+32094131608783822904581573579218183269787798627137187952457613070123^4+71030960841978946582841584808283274935807106386341917037305352370068^4=338*16739106169153734580768467570691337752935995795663937245717976794123^4
33017293667746311534136349867276879468867203935612491595409165572311^4+121046711108408304924529665080025348365792081696890300517540723010037^4+168345630985561947100372263096686425245532197169252213776784047138452^4=338*41669618230501158761044410262266083852139339942313070237356671554463^4
496763334501449423682534894785810480350037829008727967528337089295183^4+1069483421379084132507771780673365151660010761380657174159422201246219^4+2518446724648250966056412481181077783903161814966687506850325980002204^4=338*592294079196588002328582755871581473634748915077320892397073741238781^4
29849238939471830325479921749967434959090380666922222403002804248730357981^4+76564042742820064994641692849701831957190829698529360227872336014580354879^4+98378708135647450599615828485567341964982139543728518053268390242874784676^4=338*24846984390673675020871005495197567852068228956654091668728708267979178897^4
6039797270425054225654811113588126706667381256693467202026895225683944414963^4+8117323364912908684817314180638273539804371961075079460073045996673080345377^4+9180090993057704830671181929710659075445496578928187901106810991849166199668^4=338*2479456481540097626676617343443155903055896915600204292354808247886162610471^4
6626114767268163756765567294686911350330741344445242458959816824479635769039325747^4+9239570531886260346375576934821110229825451447769769322120810317871950623334085969^4+10460302725733027909952577002691364306338855519915783128226075481008271792016322044^4=338*2813798842277668027129657308851261593890265715473925880605587694274921447407765169^4
16589836196498233195536748212539551224371921384349503082739760497759882067101236771^4+28097821436027652535640426154084983905195280598056593250405931461931878207737769473^4+32657837551312034171575180950140404393278063323471860665979859517663355448193232996^4=338*8585597970822625141319462989209928993486202992800353486501224527618544858869290929^4
136293619926634734394646014641402856827780816432433429681123349199970016115268177379933339543^4+3802304370085072582160645865962803531960935042141637815255616845931504543514966277292539258251^4+6653259152758365711748014209934977022676000870768110020785157661361763928565075064184691238612^4=338*1591512288661048513709626155723201331043998677360718888743652354507068167038645619390875140543^4
159692740512805990093670293609198366544880788924125405185955601081950603001444297761927437891710743^4+1398684861559578981377745593317606675997802654605615173544402656660797993613617279729966723782122411^4+2592647955214950821476454146448307215490329699005302210216930616172588910468915976453722412272709708^4=338*617083100193719421401666476387384885420600633342054704081612047084719795572029233245128065040132943^4
7159928412411437472277216526389105117611949473378694451365981304156970167651474523975571366819998829^4+7926390305286534714770572026342950212316030207675315019798373613843498263950864682393465110853664207^4+9206375608234033299174367485693806913432479960552896409732876156480159265298892959631047400327850044^4=338*2525916371260874896828672015743546556045493239354866790129387605490454411573938318041828751914116529^4
67172129265288879708715755662651790807307143987971145980026674280850336398488821921815130251486392099^4+107284320596710099309480945133161547689818325365477590792324268599979012562026033790205323254125397721^4+123353599845764730638751871228260967662891291031279988051685597080057795468788713641912417250257470564^4=338*32655584622123537366495340316231173178188853013925291461009403349755541501473266485150897752340442517^4
1886035513264027857538746021531908335995843400696221247732954502684038905449225111637760713155280424373^4+2664899306195573023091178552253084356329903696865155577992252929654502078872454579046267358018035963873^4+7263413556394705714838742188118635915989778544979418288877860537398520979957343016899004677478099233748^4=338*1703511492518834751238891065363131771777528297335466779088789380500107912998179811994898746769725596619^4
957548823969020121230038352431706057584086650959951708298390232103333240712839157294151944862054744188133^4+2762481538867713437564145721752260503901559724935755420174346607161976136334608555125757270519874982553431^4+3650555345185672407783650051862638002414691340897773036928386481318932405785303727839035097450201195703804^4=338*914763272620637802907861580502145452536216217696637086758609216779440633899781040218435677088980970756569^4
...

u=-7/13のとき

13093^4+56009^4+107168^4=338*25449^4
14895881567697871290885579522255376391623^4+38361144876287928261741267327131864412373^4+46374157319422309032589187423493949355552^4=338*11926965907088849091128946002597489225433^4
6489756539649682566724254801697567700400257667967663818923533186424701562840755264990791777716048917271257986251^4+8538910603015842115326647176784721489284221648357753738436116706267919229831675946522539734803899315568474242073^4+8983779587676953084978903886573459327496383193034836648776827559854217298631556117641443842968854767963256600352^4=338*2518762647206162807581356850533110329020882796362797122826623638003182489759172811981562066321757486093047935161^4
657824606538347138175451082388298749786959185623749700566701723146584216058998793908358724918574379505718024431732319658879075549199349935961076197992443572141086865465206712847614125043468792703739673305079315754913861^4+762314611831329414362686254591560242094941286048608415619731230151735231479914522976150548880677148131902550639224278819285252732799381483950065730789944499895706920559571367202684681029163445491791570521551605204382103^4+812391845887079650916882808542874414201833996068540801265713044418544859401715820519582345236614237795134399051095023682091396107129210877350762563203494607625759521614268706504362662241287218018336665770333436613824928^4=338*230886924781789018089752997849091035136151879135932078477399106104522372793976874754479232238858672463828755137685515121629003151705102970006172748694691229703636392081352989825878682104862294086902966426796376564453129^4
...

[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: 2026.02.01

H.Nakao