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Integer Points on A^4+1013*B^4+4052*C^4=D^4

[2026.04.23]A^4+1013*B^4+4052*C^4=D^4の整点

■正整数nに対して、Diophantine Equation
       A^4+n*B^4+4*n*C^4=D^4 ----------(1)
を満たす自明でない整数の組(A,B,C,D) (ただし A*B*C*D!=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])の方法によって、(1)を満たす整数の組(A,B,C,D)を探す。

■(1)およびD!=0より、x=B/C,y=A/C.t=D/Cとすると、
       n*(x^4+4)+y^4=t^4
       n*(x^2+2*x+2)*(x^2-2*x+2)=(t^2+y^2)*(t^2-y^2) -----(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。

ここで、ある有理数uに対して、
       n=n1*n2 ----------(3)
       n1*(x^2+2*x+2)=u*(t^2+y^2) ----------(4a)
       ±n2*(x^2-2*x+2)=(1/u)*(t^2-y^2) ----------(4b±)
を満たす有理数の組(n1,n2,x,y,t)が存在すれば、(x,y,t)が(2)を満たすことが分かる。
(4a)-u^2*(4b+)より、
       2*u*y^2=(n1-n2*u^2)*x^2+2*(n1+n2*u^2)*x+2*(n1-n2*u^2) ----------(5a+)
(4a)+u^2*(4b+)より、
       2*u*t^2=(n1+n2*u^2)*x^2+2*(n1-n2*u^2)*x+2*(n1+n2*u^2) ----------(5b+)
となる。

また、(4a)+u^2*(4b-)より、
       2*u*t^2=(n1-n2*u^2)*x^2+2*(n1+n2*u^2)*x+2*(n1-n2*u^2) ----------(5a-)
(4a)-u^2*(4b-)より、
       2*u*y^2=(n1+n2*u^2)*x^2+2*(n1-n2*u^2)*x+2*(n1+n2*u^2) ----------(5b-)
となり、それぞれ(5b+),(5a+)と一致する。
よって、u > 0, n1, n2 > 0として良い。

[pari/gpによる計算]
(06:57) gp > YY2(n1,n2,u,x)
%1 = (1/(2*u)*n1 - 1/2*u*n2)*x^2 + (1/u*n1 + u*n2)*x + (1/u*n1 - u*n2)
(06:57) gp > TT2(n1,n2,u,x)
%2 = (1/(2*u)*n1 + 1/2*u*n2)*x^2 + (1/u*n1 - u*n2)*x + (1/u*n1 + u*n2)
(06:57) gp > n1*n2*(x^4+4)+YY2(n1,n2,u,x)^2-TT2(n1,n2,u,x)^2
%3 = 0

■２次曲線(5a),(5b)は、常にnon-singularである。
２次曲線(5a)の右辺の判別式は
    4*(n1-n2*u^2)^2-4*2*(n1+n2*u^2)^2=-4*(n1^2-6*n1*n2*u^2+n2^2*u^4) となり、有理数の根を持たないので、任意の有理数uについて、non-singularである。

同様に、２次曲線(5b)の右辺の判別式は
    4*(n1+n2*u^2)^2-4*2*(n1-n2*u^2)^2=-4*(n1^2+6*n1*n2*u^2+n2^2*u^4) となり、有理数の根を持たないので、任意の有理数uについて、non-singularである。

■以下では、n=1013とする。

■有理数u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(5a+)と(5b±)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246個のuが抽出される。
[pari/gpによる計算]
> PP(1013,1,1,200);
**u= 2/17 ;  292753*x^2 - 68*y^2 + 585522*x*z + 585506*z^2
; C5a (25059/26752 : -7651933/53504 : 1)  C5b (-547/1067 : 155767/2134 : 1)
**u= 2/29 ;  851929*x^2 - 116*y^2 + 1703874*x*z + 1703858*z^2
; C5a (166551/147230 : 59407141/294460 : 1)  C5b (-16329/11147 : 2106951/22294 : 1)
**u= 2/73 ;  5398273*x^2 - 292*y^2 + 10796562*x*z + 10796546*z^2
; C5a (-17597/8412 : -3386941/16824 : 1)  C5b (-20701/22081 : 6016337/44162 : 1)
**u= 2/113 ;  12934993*x^2 - 452*y^2 + 25870002*x*z + 25869986*z^2
; C5a (-473001/190625 : 115268681/381250 : 1)  C5b (-70103/144376 : -54931859/288752 : 1)
**u= 2/117 ;  13866953*x^2 - 468*y^2 + 27733922*x*z + 27733906*z^2
; C5a (14921/6298 : 22860061/37788 : 1)  C5b (-21647/8142 : 16286851/48852 : 1)
**u= 4/9 ;  82037*x^2 - 72*y^2 + 164138*x*z + 164074*z^2
; C5a (-974/1665 : 365029/9990 : 1)  C5b (830/727 : 348061/4362 : 1)
**u= 4/41 ;  1702837*x^2 - 328*y^2 + 3405738*x*z + 3405674*z^2
; C5a (63286/23319 : -12924763/46638 : 1)  C5b (49682/62401 : 18486329/124802 : 1)
**u= 4/65 ;  4279909*x^2 - 520*y^2 + 8559882*x*z + 8559818*z^2
; C5a (-54428/18053 : -7368189/36106 : 1)  C5b (-5194/1327 : 741817/2654 : 1)
**u= 4/101 ;  10333597*x^2 - 808*y^2 + 20667258*x*z + 20667194*z^2
; C5a (-286142/173115 : 46761197/346230 : 1)  C5b (-552430/228379 : -89666693/456758 : 1)
**u= 4/113 ;  12934981*x^2 - 904*y^2 + 25870026*x*z + 25869962*z^2
; C5a (920446/546391 : -374477373/1092782 : 1)  C5b (4485902/1969959 : -538262681/1313306 : 1)
**u= 4/125 ;  15828109*x^2 - 1000*y^2 + 31656282*x*z + 31656218*z^2
; C5a (-5002/1161 : -5048279/11610 : 1)  C5b (-654152/1399467 : 132986285/932978 : 1)
**u= 4/181 ;  33186877*x^2 - 1448*y^2 + 66373818*x*z + 66373754*z^2
; C5a (-72344/101015 : 31793529/202030 : 1)  C5b (14656/286473 : 41947991/190982 : 1)
**u= 8/5 ;  25261*x^2 - 80*y^2 + 50778*x*z + 50522*z^2
; C5a (-4227/6431 : -481765/25724 : 1)  C5b (-2297/3177 : 78569/4236 : 1)
**u= 8/13 ;  171133*x^2 - 208*y^2 + 342522*x*z + 342266*z^2
; C5a (-1787/1299 : 159067/5196 : 1)  C5b (-1985/77 : -219181/308 : 1)
**u= 8/25 ;  633061*x^2 - 400*y^2 + 1266378*x*z + 1266122*z^2
; C5a (2773/770 : 262263/1400 : 1)  C5b (-263/525 : 155643/3500 : 1)
**u= 8/41 ;  1702789*x^2 - 656*y^2 + 3405834*x*z + 3405578*z^2
; C5a (-15439/5381 : -2324541/21524 : 1)  C5b (9565/142 : -1978499/568 : 1)
**u= 8/49 ;  2432149*x^2 - 784*y^2 + 4864554*x*z + 4864298*z^2
; C5a (347/162 : 833047/4536 : 1)  C5b (-4781/4094 : -6474563/114632 : 1)
**u= 8/53 ;  2845453*x^2 - 848*y^2 + 5691162*x*z + 5690906*z^2
; C5a (169156059/83464297 : -61646322629/333857188 : 1)  C5b (-36355/19911 : 1994631/26548 : 1)
**u= 8/81 ;  6646229*x^2 - 1296*y^2 + 13292714*x*z + 13292458*z^2
; C5a (-1043/1454 : -3895279/52344 : 1)  C5b (-47815/24953 : -87249829/898308 : 1)
**u= 8/97 ;  9531253*x^2 - 1552*y^2 + 19062762*x*z + 19062506*z^2
; C5a (16544407/22405430 : 14085373719/89621720 : 1)  C5b (-481835/394473 : 42217479/525964 : 1)
**u= 8/117 ;  13866893*x^2 - 1872*y^2 + 27734042*x*z + 27733786*z^2
; C5a (-44949/19828 : 33052919/237936 : 1)  C5b (407855/604973 : -1218455371/7259676 : 1)
**u= 8/149 ;  22489549*x^2 - 2384*y^2 + 44979354*x*z + 44979098*z^2
; C5a (651243/913156 : 703742717/3652624 : 1)  C5b (-25135/14507 : 78503/652 : 1)
**u= 8/153 ;  23713253*x^2 - 2448*y^2 + 47426762*x*z + 47426506*z^2
; C5a (66443/32315 : 5335847/16860 : 1)  C5b (-15393/1678 : -16319047/20136 : 1)
**u= 8/173 ;  30318013*x^2 - 2768*y^2 + 60636282*x*z + 60636026*z^2
; C5a (674357/1719655 : 1233957303/6878620 : 1)  C5b (677045/683259 : -212419761/911012 : 1)
**u= 8/181 ;  33186829*x^2 - 2896*y^2 + 66373914*x*z + 66373658*z^2
; C5a (398773/100755 : 218204111/403020 : 1)  C5b (119281/113831 : 111082957/455324 : 1)
**u= 10 ;  913*x^2 - 20*y^2 + 2226*x*z + 1826*z^2
; C5a (3/34 : -685/68 : 1)  C5b (-113/31 : -1411/62 : 1)
**u= 10/89 ;  8023873*x^2 - 1780*y^2 + 16048146*x*z + 16047746*z^2
; C5a (8892907/78788451 : -15828940591/157576902 : 1)  C5b (-85687/76534 : -10350649/153068 : 1)
**u= 10/117 ;  13866857*x^2 - 2340*y^2 + 27734114*x*z + 27733714*z^2
; C5a (70699/16824 : -41165593/100944 : 1)  C5b (4089/5719 : -5244055/34314 : 1)
**u= 10/137 ;  19012897*x^2 - 2740*y^2 + 38026194*x*z + 38025794*z^2
; C5a (-5431019/6509081 : 1099186149/13018162 : 1)  C5b (-290393/85422 : 12331895/56948 : 1)
**u= 10/169 ;  28932193*x^2 - 3380*y^2 + 57864786*x*z + 57864386*z^2
; C5a (10423/419 : -26099925/10894 : 1)  C5b (-1421/12522 : -13418083/108524 : 1)
**u= 16 ;  757*x^2 - 32*y^2 + 2538*x*z + 1514*z^2
; C5a (-52/67 : -21/268 : 1)  C5b (10/97 : -3569/388 : 1)
**u= 16/13 ;  170941*x^2 - 416*y^2 + 342906*x*z + 341882*z^2
; C5a (584488/110485 : 57081303/441940 : 1)  C5b (192/617 : 82593/2468 : 1)
**u= 16/45 ;  2051069*x^2 - 1440*y^2 + 4103162*x*z + 4102138*z^2
; C5a (798/6397 : -4360253/76764 : 1)  C5b (-32336/33181 : -15037655/398172 : 1)
**u= 16/101 ;  10333357*x^2 - 3232*y^2 + 20667738*x*z + 20666714*z^2
; C5a (-4882/7085 : 1678077/28340 : 1)  C5b (-298374/178759 : 6950199/102148 : 1)
**u= 16/113 ;  12934741*x^2 - 3616*y^2 + 25870506*x*z + 25869482*z^2
; C5a (2886922/368929 : -783901047/1475716 : 1)  C5b (-73084/16827 : -4682757/22436 : 1)
**u= 16/169 ;  28932037*x^2 - 5408*y^2 + 57865098*x*z + 57864074*z^2
; C5a (7894/1295 : 35295081/67340 : 1)  C5b (-1966/1961 : 7458733/101972 : 1)
**u= 16/185 ;  34669669*x^2 - 5920*y^2 + 69340362*x*z + 69339338*z^2
; C5a (-2012/6537 : -2433653/26148 : 1)  C5b (-519506/1551023 : -570196601/6204092 : 1)
**u= 18/13 ;  170873*x^2 - 468*y^2 + 343042*x*z + 341746*z^2
; C5a (79381/61549 : 17644787/369294 : 1)  C5b (-327/449 : 53581/2694 : 1)
**u= 18/17 ;  292433*x^2 - 612*y^2 + 586162*x*z + 584866*z^2
; C5a (-21883/1536 : 2675749/9216 : 1)  C5b (-4117/472 : 482743/2832 : 1)
**u= 18/41 ;  1702529*x^2 - 1476*y^2 + 3406354*x*z + 3405058*z^2
; C5a (-12929/5301 : 1892329/31806 : 1)  C5b (-19745/3704 : 3355783/22224 : 1)
**u= 18/65 ;  4279601*x^2 - 2340*y^2 + 8560498*x*z + 8559202*z^2
; C5a (-26063/18144 : -5078807/108864 : 1)  C5b (-5303/4488 : -1170715/26928 : 1)
**u= 18/85 ;  7318601*x^2 - 3060*y^2 + 14638498*x*z + 14637202*z^2
; C5a (-124563/18094 : 31688599/108564 : 1)  C5b (-1699/1469 : 436363/8814 : 1)
**u= 18/97 ;  9530993*x^2 - 3492*y^2 + 19063282*x*z + 19061986*z^2
; C5a (831363/974132 : -643080313/5844792 : 1)  C5b (-4749/33608 : 13886329/201648 : 1)
**u= 18/185 ;  34669601*x^2 - 6660*y^2 + 69340498*x*z + 69339202*z^2
; C5a (-2830591/1316863 : 868542331/7901178 : 1)  C5b (-3088879/2117419 : -1008533251/12704514 : 1)
**u= 20/9 ;  81653*x^2 - 360*y^2 + 164906*x*z + 163306*z^2
; C5a (-1008/541 : 63901/3246 : 1)  C5b (896/1671 : 277697/10026 : 1)
**u= 20/17 ;  292357*x^2 - 680*y^2 + 586314*x*z + 584714*z^2
; C5a (-5034/9421 : 430451/18842 : 1)  C5b (-1846/1647 : 23033/1098 : 1)
**u= 20/89 ;  8023573*x^2 - 3560*y^2 + 16048746*x*z + 16047146*z^2
; C5a (-4009256/3192977 : -312882327/6385954 : 1)  C5b (-58742/65359 : 891191/18674 : 1)
**u= 20/101 ;  10333213*x^2 - 4040*y^2 + 20668026*x*z + 20666426*z^2
; C5a (-199212/207739 : -21028505/415478 : 1)  C5b (-2874/14291 : -1850241/28582 : 1)
**u= 20/117 ;  13866557*x^2 - 4680*y^2 + 27734714*x*z + 27733114*z^2
; C5a (-225506/93239 : 52849913/559434 : 1)  C5b (-182186/136059 : 6703621/116622 : 1)
**u= 26/25 ;  632449*x^2 - 1300*y^2 + 1267602*x*z + 1264898*z^2
; C5a (-1013521/2141538 : 106690081/4283076 : 1)  C5b (-11285/12647 : 2813921/126470 : 1)
**u= 26/45 ;  2050649*x^2 - 2340*y^2 + 4104002*x*z + 4101298*z^2
; C5a (232481/174584 : 78682333/1047504 : 1)  C5b (4749/3628 : 1621775/21768 : 1)
**u= 26/53 ;  2844841*x^2 - 2756*y^2 + 5692386*x*z + 5689682*z^2
; C5a (-133641/118649 : 7680581/237298 : 1)  C5b (-1001369/988341 : 21186281/658894 : 1)
**u= 26/113 ;  12934321*x^2 - 5876*y^2 + 25871346*x*z + 25868642*z^2
; C5a (-6481029/8707291 : 843262399/17414582 : 1)  C5b (35941/5706 : 1314859/3804 : 1)
**u= 26/125 ;  15827449*x^2 - 6500*y^2 + 31657602*x*z + 31654898*z^2
; C5a (-31713/57868 : -31335359/578680 : 1)  C5b (219437/10019 : -113338853/100190 : 1)
**u= 26/137 ;  19012321*x^2 - 7124*y^2 + 38027346*x*z + 38024642*z^2
; C5a (-184621/123523 : -14237091/247046 : 1)  C5b (-141383995/45449934 : 3656292027/30299956 : 1)
**u= 26/145 ;  21297649*x^2 - 7540*y^2 + 42598002*x*z + 42595298*z^2
; C5a (6266297/140718 : 681194621/281436 : 1)  C5b (452027/441039 : -35291571/294026 : 1)
**u= 26/193 ;  37732561*x^2 - 10036*y^2 + 75467826*x*z + 75465122*z^2
; C5a (-137664803/156430049 : 19320440469/312860098 : 1)  C5b (-187296787/227440453 : 28324174789/454880906 : 1)
**u= 32 ;  -11*x^2 - 64*y^2 + 4074*x*z - 22*z^2
; C5a (1409/9 : 5461/72 : 1)  C5b (-67/58 : 4789/464 : 1)
**u= 32/9 ;  81029*x^2 - 576*y^2 + 166154*x*z + 162058*z^2
; C5a (-843/242 : 182147/5808 : 1)  C5b (89/293 : -138377/7032 : 1)
**u= 32/29 ;  850909*x^2 - 1856*y^2 + 1705914*x*z + 1701818*z^2
; C5a (353421/199652 : -100771399/1597216 : 1)  C5b (-3151/3599 : -623279/28792 : 1)
**u= 32/41 ;  1701829*x^2 - 2624*y^2 + 3407754*x*z + 3403658*z^2
; C5a (369059/331618 : 157972077/2652944 : 1)  C5b (1520709/2975834 : 1099010793/23806672 : 1)
**u= 32/97 ;  9530293*x^2 - 6208*y^2 + 19064682*x*z + 19060586*z^2
; C5a (1424359/2851490 : 1611008043/22811920 : 1)  C5b (150449/479062 : 247979381/3832496 : 1)
**u= 32/113 ;  12933973*x^2 - 7232*y^2 + 25872042*x*z + 25867946*z^2
; C5a (-80043/30218 : 19712411/241744 : 1)  C5b (-5143319/15686233 : -6394977773/125489864 : 1)
**u= 32/121 ;  14830309*x^2 - 7744*y^2 + 29664714*x*z + 29660618*z^2
; C5a (7433/3565 : 44524209/313720 : 1)  C5b (-42073/3319 : -149800771/292072 : 1)
**u= 32/125 ;  15827101*x^2 - 8000*y^2 + 31658298*x*z + 31654202*z^2
; C5a (-10429/36892 : 80774979/1475680 : 1)  C5b (2231/4569 : 4858769/60920 : 1)
**u= 32/185 ;  34668901*x^2 - 11840*y^2 + 69341898*x*z + 69337802*z^2
; C5a (100169/122986 : 110303859/983888 : 1)  C5b (57381/132734 : -100376301/1061872 : 1)
**u= 34 ;  -143*x^2 - 68*y^2 + 4338*x*z - 286*z^2
; C5a (271/704 : -573/128 : 1)  C5b (23/9 : 97/6 : 1)
**u= 34/9 ;  80897*x^2 - 612*y^2 + 166418*x*z + 161794*z^2
; C5a (871/2625 : 302609/15750 : 1)  C5b (-513295/616183 : -44679419/3697098 : 1)
**u= 34/29 ;  850777*x^2 - 1972*y^2 + 1706178*x*z + 1701554*z^2
; C5a (-32873/73371 : -3478181/146742 : 1)  C5b (139157/28561 : 7074557/57122 : 1)
**u= 34/65 ;  4278769*x^2 - 4420*y^2 + 8562162*x*z + 8557538*z^2
; C5a (-67033/92349 : -5956435/184698 : 1)  C5b (-115853/154644 : -472743/14728 : 1)
**u= 34/73 ;  5397121*x^2 - 4964*y^2 + 10798866*x*z + 10794242*z^2
; C5a (-189029/336812 : -24250899/673624 : 1)  C5b (-24563/43744 : 3151237/87488 : 1)
**u= 34/117 ;  13865801*x^2 - 7956*y^2 + 27736226*x*z + 27731602*z^2
; C5a (-213937/97031 : 38049433/582186 : 1)  C5b (261377/147937 : -109022083/887622 : 1)
**u= 34/145 ;  21297169*x^2 - 9860*y^2 + 42598962*x*z + 42594338*z^2
; C5a (12891/30817 : 4971047/61634 : 1)  C5b (998641/1445587 : 263962433/2891174 : 1)
**u= 34/173 ;  30316921*x^2 - 11764*y^2 + 60638466*x*z + 60633842*z^2
; C5a (-11350621/10740234 : 1092129427/21480468 : 1)  C5b (-893755/701506 : 73859701/1403012 : 1)
**u= 36/49 ;  2430917*x^2 - 3528*y^2 + 4867018*x*z + 4861834*z^2
; C5a (-7684/3499 : -6008243/146958 : 1)  C5b (-164/1533 : 2267273/64386 : 1)
**u= 36/149 ;  22488317*x^2 - 10728*y^2 + 44981818*x*z + 44976634*z^2
; C5a (375534/401437 : -240250477/2408622 : 1)  C5b (-9144/8969 : -2464751/53814 : 1)
**u= 40/149 ;  22488013*x^2 - 11920*y^2 + 44982426*x*z + 44976026*z^2
; C5a (-1002917/2958738 : -616186543/11834952 : 1)  C5b (170529/11402 : 31672521/45608 : 1)
**u= 40/153 ;  23711717*x^2 - 12240*y^2 + 47429834*x*z + 47423434*z^2
; C5a (-1743/10307 : 7077677/123684 : 1)  C5b (-9496021/3984132 : 3592733377/47809584 : 1)
**u= 40/193 ;  37731637*x^2 - 15440*y^2 + 75469674*x*z + 75463274*z^2
; C5a (-447159/196411 : 62977765/785644 : 1)  C5b (-1354427/2144859 : -150680961/2859812 : 1)
**u= 50/9 ;  79553*x^2 - 900*y^2 + 169106*x*z + 159106*z^2
; C5a (-39/32 : -8537/960 : 1)  C5b (-31/563 : 225623/16890 : 1)
**u= 50/81 ;  6643793*x^2 - 8100*y^2 + 13297586*x*z + 13287586*z^2
; C5a (-61497/51881 : 135886409/4669290 : 1)  C5b (-5185/7352 : 19773293/661680 : 1)
**u= 50/109 ;  12032953*x^2 - 10900*y^2 + 24075906*x*z + 24065906*z^2
; C5a (2946203/2846159 : -428901459/5692318 : 1)  C5b (-69313/73399 : 24439643/733990 : 1)
**u= 50/173 ;  30315577*x^2 - 17300*y^2 + 60641154*x*z + 60631154*z^2
; C5a (-5403/25541 : 2723009/51082 : 1)  C5b (249311/84059 : 143927713/840590 : 1)
**u= 52/41 ;  1700149*x^2 - 4264*y^2 + 3411114*x*z + 3400298*z^2
; C5a (-1428142/816611 : 40597569/1633222 : 1)  C5b (-207562/158053 : 2819/134 : 1)
**u= 52/49 ;  2429509*x^2 - 5096*y^2 + 4869834*x*z + 4859018*z^2
; C5a (-2342/2411 : -735711/33754 : 1)  C5b (-5448/2629 : 1182141/36806 : 1)
**u= 52/53 ;  2842813*x^2 - 5512*y^2 + 5696442*x*z + 5685626*z^2
; C5a (-3341906/2066371 : -110050233/4132742 : 1)  C5b (-13222/1017 : 185633/678 : 1)
**u= 52/61 ;  3766669*x^2 - 6344*y^2 + 7544154*x*z + 7533338*z^2
; C5a (-298716/316585 : -15431953/633170 : 1)  C5b (-191118/432149 : -24143349/864298 : 1)
**u= 52/89 ;  8021269*x^2 - 9256*y^2 + 16053354*x*z + 16042538*z^2
; C5a (-51688816/28824943 : -2164551729/57649886 : 1)  C5b (-333656/2009 : -19533139/4018 : 1)
**u= 52/137 ;  19010293*x^2 - 14248*y^2 + 38031402*x*z + 38020586*z^2
; C5a (-233626/143345 : 12371787/286690 : 1)  C5b (-57352/323467 : -30605297/646934 : 1)
**u= 52/181 ;  33184189*x^2 - 18824*y^2 + 66379194*x*z + 66368378*z^2
; C5a (-1024038/1222477 : 103984883/2444954 : 1)  C5b (707884/133279 : -71520443/266558 : 1)
**u= 58/5 ;  21961*x^2 - 580*y^2 + 57378*x*z + 43922*z^2
; C5a (-1909/2157 : -18217/4314 : 1)  C5b (3469/6131 : -153941/12262 : 1)
**u= 58/17 ;  289393*x^2 - 1972*y^2 + 592242*x*z + 578786*z^2
; C5a (-31547/69874 : -1915227/139748 : 1)  C5b (-41693/4646 : -917951/9292 : 1)
**u= 58/41 ;  1699489*x^2 - 4756*y^2 + 3412434*x*z + 3398978*z^2
; C5a (-135127/118290 : 4497157/236580 : 1)  C5b (213305/161614 : -15453179/323228 : 1)
**u= 58/85 ;  7315561*x^2 - 9860*y^2 + 14644578*x*z + 14631122*z^2
; C5a (-2838051/5621663 : -341613091/11243326 : 1)  C5b (-80173/58269 : -1132175/38846 : 1)
**u= 58/113 ;  12931633*x^2 - 13108*y^2 + 25876722*x*z + 25863266*z^2
; C5a (-548807/208870 : 25053891/417740 : 1)  C5b (-231197/25079 : -13047847/50158 : 1)
**u= 58/137 ;  19009633*x^2 - 15892*y^2 + 38032722*x*z + 38019266*z^2
; C5a (-551231/162826 : 29126667/325652 : 1)  C5b (-84281341/88560207 : 2045379769/59040138 : 1)
**u= 58/169 ;  28928929*x^2 - 19604*y^2 + 57871314*x*z + 57857858*z^2
; C5a (-12873/73655 : 95376733/1915030 : 1)  C5b (29005/214 : 2245177/428 : 1)
**u= 58/173 ;  30314713*x^2 - 20068*y^2 + 60642882*x*z + 60629426*z^2
; C5a (-8937323/7789156 : -611863521/15578312 : 1)  C5b (-503815/350568 : 9917279/233712 : 1)
**u= 58/193 ;  37729873*x^2 - 22388*y^2 + 75473202*x*z + 75459746*z^2
; C5a (-229722709/262166134 : 21685779429/524332268 : 1)  C5b (-214391/764241 : 25769827/509494 : 1)
**u= 64/5 ;  21229*x^2 - 640*y^2 + 58842*x*z + 42458*z^2
; C5a (-434/149 : -10659/1192 : 1)  C5b (-486/583 : -38625/4664 : 1)
**u= 64/13 ;  167101*x^2 - 1664*y^2 + 350586*x*z + 334202*z^2
; C5a (-1104/211 : 72557/1688 : 1)  C5b (-11266/13289 : 1145671/106312 : 1)
**u= 64/149 ;  22485517*x^2 - 19072*y^2 + 44987418*x*z + 44971034*z^2
; C5a (374726/22469 : -109282161/179752 : 1)  C5b (370520/31887 : 36967019/85032 : 1)
**u= 64/153 ;  23709221*x^2 - 19584*y^2 + 47434826*x*z + 47418442*z^2
; C5a (133186/23021 : -131857229/552504 : 1)  C5b (-238922/522723 : 496838479/12545352 : 1)
**u= 68/9 ;  77429*x^2 - 1224*y^2 + 173354*x*z + 154858*z^2
; C5a (-23746/12343 : 695429/74058 : 1)  C5b (-1580/669 : -61901/4014 : 1)
**u= 68/73 ;  5393653*x^2 - 9928*y^2 + 10805802*x*z + 10787306*z^2
; C5a (-418/835 : -43479/1670 : 1)  C5b (-495908/118361 : 18472223/236722 : 1)
**u= 68/89 ;  8019349*x^2 - 12104*y^2 + 16057194*x*z + 16038698*z^2
; C5a (360582/128779 : 26055251/257558 : 1)  C5b (113180/320951 : 27805253/641902 : 1)
**u= 68/101 ;  10328989*x^2 - 13736*y^2 + 20676474*x*z + 20657978*z^2
; C5a (-78648/38197 : -3048611/76394 : 1)  C5b (-239256/589529 : -37635471/1179058 : 1)
**u= 72/29 ;  846749*x^2 - 4176*y^2 + 1714234*x*z + 1693498*z^2
; C5a (1942187/2997912 : 989521379/35974944 : 1)  C5b (4001/2427 : 1178291/29124 : 1)
**u= 72/101 ;  10328429*x^2 - 14544*y^2 + 20677594*x*z + 20656858*z^2
; C5a (1028261/1048167 : 743961107/12578004 : 1)  C5b (-64385/28016 : 14700797/336192 : 1)
**u= 72/113 ;  12929813*x^2 - 16272*y^2 + 25880362*x*z + 25859626*z^2
; C5a (926423/932359 : -703538003/11188308 : 1)  C5b (40421/106338 : 61323437/1276056 : 1)
**u= 72/145 ;  21293141*x^2 - 20880*y^2 + 42607018*x*z + 42586282*z^2
; C5a (-24437349/102679942 : -49466096779/1232159304 : 1)  C5b (-15611/586 : -5763611/7032 : 1)
**u= 72/169 ;  28927109*x^2 - 24336*y^2 + 57874954*x*z + 57854218*z^2
; C5a (-32037/21226 : 9850883/254712 : 1)  C5b (-5013/26782 : -14282047/321384 : 1)
**u= 72/185 ;  34664741*x^2 - 26640*y^2 + 69350218*x*z + 69329482*z^2
; C5a (10621/199622 : -125496899/2395464 : 1)  C5b (-36750793/42752323 : 18695184637/513027876 : 1)
**u= 74/45 ;  2045849*x^2 - 6660*y^2 + 4113602*x*z + 4091698*z^2
; C5a (-8329/5009 : 628025/30054 : 1)  C5b (-156813/124844 : -13673461/749064 : 1)
**u= 74/49 ;  2426737*x^2 - 7252*y^2 + 4875378*x*z + 4853474*z^2
; C5a (-9893/13850 : -3677907/193900 : 1)  C5b (-17341/27151 : 7428787/380114 : 1)
**u= 74/65 ;  4274449*x^2 - 9620*y^2 + 8570802*x*z + 8548898*z^2
; C5a (-16149/274174 : 15871087/548348 : 1)  C5b (-8197/5426 : -257969/10852 : 1)
**u= 74/81 ;  6640817*x^2 - 11988*y^2 + 13303538*x*z + 13281634*z^2
; C5a (102169/30962 : 57922691/557316 : 1)  C5b (10937/9638 : -9630299/173484 : 1)
**u= 74/97 ;  9525841*x^2 - 14356*y^2 + 19073586*x*z + 19051682*z^2
; C5a (118103/5350 : 6366411/10700 : 1)  C5b (-1894979/1635521 : -85473347/3271042 : 1)
**u= 74/197 ;  39308041*x^2 - 29156*y^2 + 78637986*x*z + 78616082*z^2
; C5a (-2615199/2556880 : 187761383/5113760 : 1)  C5b (-1375421/1054228 : 11567069/301208 : 1)
**u= 80/61 ;  3762973*x^2 - 9760*y^2 + 7551546*x*z + 7525946*z^2
; C5a (-102194/51823 : -5656551/207292 : 1)  C5b (-54404/53213 : 4202075/212852 : 1)
**u= 80/117 ;  13860557*x^2 - 18720*y^2 + 27746714*x*z + 27721114*z^2
; C5a (19378/38633 : 22761433/463596 : 1)  C5b (-183382/125373 : 45178267/1504476 : 1)
**u= 80/137 ;  19006597*x^2 - 21920*y^2 + 38038794*x*z + 38013194*z^2
; C5a (107672/645217 : 116794329/2580868 : 1)  C5b (-272354/211081 : 25918043/844324 : 1)
**u= 80/173 ;  30311677*x^2 - 27680*y^2 + 60648954*x*z + 60623354*z^2
; C5a (-3374418/2177567 : 328742369/8710268 : 1)  C5b (-600046/610469 : 80868721/2441876 : 1)
**u= 80/197 ;  39307117*x^2 - 31520*y^2 + 78639834*x*z + 78614234*z^2
; C5a (-4632978/3119611 : 489581861/12478444 : 1)  C5b (-9148/202287 : 13171089/269716 : 1)
**u= 82/9 ;  75329*x^2 - 1476*y^2 + 177554*x*z + 150658*z^2
; C5a (-14179/6156 : 361429/36936 : 1)  C5b (1981/872 : -134677/5232 : 1)
**u= 82/17 ;  286033*x^2 - 2788*y^2 + 598962*x*z + 572066*z^2
; C5a (-79153/42425 : -1078251/84850 : 1)  C5b (-3703/17203 : 455987/34406 : 1)
**u= 82/65 ;  4273201*x^2 - 10660*y^2 + 8573298*x*z + 8546402*z^2
; C5a (-1246507/73392 : -47057231/146784 : 1)  C5b (-233803/18079 : -8684633/36158 : 1)
**u= 82/81 ;  6639569*x^2 - 13284*y^2 + 13306034*x*z + 13279138*z^2
; C5a (29917/1500 : -12658349/27000 : 1)  C5b (-28707/47431 : 20563039/853758 : 1)
**u= 82/85 ;  7312201*x^2 - 13940*y^2 + 14651298*x*z + 14624402*z^2
; C5a (2040853/1552121 : 179347977/3104242 : 1)  C5b (-761659/545474 : -26961011/1090948 : 1)
**u= 82/109 ;  12028729*x^2 - 17876*y^2 + 24084354*x*z + 24057458*z^2
; C5a (267371/16030 : -14727381/32060 : 1)  C5b (35257/32569 : -3904807/65138 : 1)
**u= 82/125 ;  15821401*x^2 - 20500*y^2 + 31669698*x*z + 31642802*z^2
; C5a (-360403/577474 : -171307239/5774740 : 1)  C5b (-121081/185206 : 54498673/1852060 : 1)
**u= 82/153 ;  23706593*x^2 - 25092*y^2 + 47440082*x*z + 47413186*z^2
; C5a (119620361/111188479 : 47253918139/667130874 : 1)  C5b (2503/88632 : 23451737/531792 : 1)
**u= 82/193 ;  37726513*x^2 - 31652*y^2 + 75479922*x*z + 75453026*z^2
; C5a (311271/22760 : -23118259/45520 : 1)  C5b (-8028929/3005704 : 404368607/6011408 : 1)
**u= 90/73 ;  5390177*x^2 - 13140*y^2 + 10812754*x*z + 10780354*z^2
; C5a (-24623/17577 : -2292859/105462 : 1)  C5b (-117667/66678 : -10249843/400068 : 1)
**u= 90/89 ;  8015873*x^2 - 16020*y^2 + 16064146*x*z + 16031746*z^2
; C5a (-47477/43558 : -5856835/261348 : 1)  C5b (8369/542 : 1199257/3252 : 1)
**u= 90/113 ;  12926897*x^2 - 20340*y^2 + 25886194*x*z + 25853794*z^2
; C5a (-1600849/2220626 : -348435697/13323756 : 1)  C5b (-118799/113026 : -17151737/678156 : 1)
**u= 90/121 ;  14823233*x^2 - 21780*y^2 + 29678866*x*z + 29646466*z^2
; C5a (-168947/2286434 : -5365515455/150904644 : 1)  C5b (36083/6454 : -10587053/60852 : 1)
**u= 90/197 ;  39305417*x^2 - 35460*y^2 + 78643234*x*z + 78610834*z^2
; C5a (-147691/45419 : 22348933/272514 : 1)  C5b (-20881/212669 : -57219311/1276014 : 1)
**u= 98/37 ;  1377193*x^2 - 7252*y^2 + 2792802*x*z + 2754386*z^2
; C5a (-5407/11582 : 2519373/162148 : 1)  C5b (569/10621 : -2996131/148694 : 1)
**u= 98/45 ;  2041721*x^2 - 8820*y^2 + 4121858*x*z + 4083442*z^2
; C5a (251/1527 : 1498661/64134 : 1)  C5b (-957/8882 : -7646581/373044 : 1)
**u= 98/85 ;  7309321*x^2 - 16660*y^2 + 14657058*x*z + 14618642*z^2
; C5a (-7141/25098 : -9045145/351372 : 1)  C5b (-262921/390123 : -4469593/202286 : 1)
**u= 98/89 ;  8014369*x^2 - 17444*y^2 + 16067154*x*z + 16028738*z^2
; C5a (323653723971/314463350836 : 213579092205197/4402486911704 : 1)  C5b (5461/442336 : -189104221/6192704 : 1)
**u= 98/117 ;  13857353*x^2 - 22932*y^2 + 27753122*x*z + 27714706*z^2
; C5a (-43779/19009 : 32198297/798378 : 1)  C5b (34715/52002 : 104441831/2184084 : 1)
**u= 98/149 ;  22480009*x^2 - 29204*y^2 + 44998434*x*z + 44960018*z^2
; C5a (-79967/188626 : 84530703/2640764 : 1)  C5b (-8257/35603 : -17447083/498442 : 1)
**u= 98/157 ;  24959833*x^2 - 30772*y^2 + 49958082*x*z + 49919666*z^2
; C5a (-9919/17334 : 7514471/242676 : 1)  C5b (-58751921/28300706 : -16594223341/396209884 : 1)
**u= 98/169 ;  28922689*x^2 - 33124*y^2 + 57883794*x*z + 57845378*z^2
; C5a (-231559/120395 : 880665867/21911890 : 1)  C5b (22759/17364 : 78388851/1053416 : 1)
**u= 98/197 ;  39303913*x^2 - 38612*y^2 + 78646242*x*z + 78607826*z^2
; C5a (-2031657/393242 : -752507843/5505388 : 1)  C5b (406885/1684618 : 1199798423/23584652 : 1)
**u= 100/17 ;  282757*x^2 - 3400*y^2 + 605514*x*z + 565514*z^2
; C5a (1826/3477 : -584707/34770 : 1)  C5b (-3532/3751 : 375901/37510 : 1)
**u= 100/29 ;  841933*x^2 - 5800*y^2 + 1723866*x*z + 1683866*z^2
; C5a (-24136/3611 : 2498889/36110 : 1)  C5b (-7088/183 : 280861/610 : 1)
**u= 100/73 ;  5388277*x^2 - 14600*y^2 + 10816554*x*z + 10776554*z^2
; C5a (2007554/641333 : 523915311/6413330 : 1)  C5b (41374/3579 : 578463/2386 : 1)
**u= 100/157 ;  24959437*x^2 - 31400*y^2 + 49958874*x*z + 49918874*z^2
; C5a (937118/2800741 : 1316980689/28007410 : 1)  C5b (537822/108041 : 36935397/216082 : 1)
**u= 104/157 ;  24958621*x^2 - 32656*y^2 + 49960506*x*z + 49917242*z^2
; C5a (-9783677/12633490 : -1431238713/50533960 : 1)  C5b (-4767/1633 : 391173/6532 : 1)
**u= 106/5 ;  14089*x^2 - 1060*y^2 + 73122*x*z + 28178*z^2
; C5a (167/1148 : 13947/2296 : 1)  C5b (-3253/1264 : -38257/2528 : 1)
**u= 106/13 ;  159961*x^2 - 2756*y^2 + 364866*x*z + 319922*z^2
; C5a (42761/69815 : 2066067/139630 : 1)  C5b (109925711/331189856 : -8842484183/662379712 : 1)
**u= 106/25 ;  621889*x^2 - 5300*y^2 + 1288722*x*z + 1243778*z^2
; C5a (1745/154 : 206931/1540 : 1)  C5b (491/4818 : -263077/16060 : 1)
**u= 106/37 ;  1375561*x^2 - 7844*y^2 + 2796066*x*z + 2751122*z^2
; C5a (45657/272927 : 11123479/545854 : 1)  C5b (579515/519668 : 32356997/1039336 : 1)
**u= 106/97 ;  9520081*x^2 - 20564*y^2 + 19085106*x*z + 19040162*z^2
; C5a (413629/131554 : 24143793/263108 : 1)  C5b (-1019405/1084771 : 46923361/2169542 : 1)
**u= 106/109 ;  12024217*x^2 - 23108*y^2 + 24093378*x*z + 24048434*z^2
; C5a (-187393/153388 : -7152201/306776 : 1)  C5b (-798867/365464 : -25932339/730928 : 1)
**u= 106/121 ;  14820097*x^2 - 25652*y^2 + 29685138*x*z + 29640194*z^2
; C5a (-4980523/4497505 : 2387961759/98945110 : 1)  C5b (-100729/118538 : 9073373/372548 : 1)
**u= 106/137 ;  19001761*x^2 - 29044*y^2 + 38048466*x*z + 38003522*z^2
; C5a (-80231203/89967070 : 4624419807/179934140 : 1)  C5b (73909/536513 : 41595679/1073026 : 1)
**u= 106/181 ;  33175657*x^2 - 38372*y^2 + 66396258*x*z + 66351314*z^2
; C5a (277063/20704 : 17553969/41408 : 1)  C5b (-797417/463869 : -3737353/103082 : 1)
**u= 116/25 ;  619669*x^2 - 5800*y^2 + 1293162*x*z + 1239338*z^2
; C5a (5566/13 : 576723/130 : 1)  C5b (-7694/21243 : 896217/70810 : 1)
**u= 116/101 ;  10320157*x^2 - 23432*y^2 + 20694138*x*z + 20640314*z^2
; C5a (126478/121487 : -11595897/242974 : 1)  C5b (261220/81859 : 14816891/163718 : 1)
**u= 116/157 ;  24955981*x^2 - 36424*y^2 + 49965786*x*z + 49911962*z^2
; C5a (-2734868/10439685 : 679128887/20879370 : 1)  C5b (-1900/596839 : 44140699/1193678 : 1)
**u= 116/181 ;  33173437*x^2 - 41992*y^2 + 66400698*x*z + 66346874*z^2
; C5a (-198154412/113895845 : -7956796761/227791690 : 1)  C5b (127766/337003 : -32281469/674006 : 1)
**u= 116/185 ;  34656469*x^2 - 42920*y^2 + 69366762*x*z + 69312938*z^2
; C5a (-68262/69071 : 3922693/138142 : 1)  C5b (-528912/417371 : 24584499/834742 : 1)
**u= 122/17 ;  277873*x^2 - 4148*y^2 + 615282*x*z + 555746*z^2
; C5a (7147/3466 : 186609/6932 : 1)  C5b (439867/29626 : -8056307/59252 : 1)
**u= 122/25 ;  618241*x^2 - 6100*y^2 + 1296018*x*z + 1236482*z^2
; C5a (-1507423/1571733 : 30177275/3143466 : 1)  C5b (-943/229 : 15727/458 : 1)
**u= 122/81 ;  6631409*x^2 - 19764*y^2 + 13322354*x*z + 13262818*z^2
; C5a (-3796461/3633794 : -1193687671/65408292 : 1)  C5b (94345/66561 : 57488473/1198098 : 1)
**u= 122/85 ;  7304041*x^2 - 20740*y^2 + 14667618*x*z + 14608082*z^2
; C5a (-41129/23044 : -1094487/46088 : 1)  C5b (285019/545361 : -12446259/363574 : 1)
**u= 122/109 ;  12020569*x^2 - 26596*y^2 + 24100674*x*z + 24041138*z^2
; C5a (-470213/623265 : -27239887/1246530 : 1)  C5b (-1327153/915649 : 42863477/1831298 : 1)
**u= 122/113 ;  12920113*x^2 - 27572*y^2 + 25899762*x*z + 25840226*z^2
; C5a (-4557/10994 : -551167/21988 : 1)  C5b (-17682897/17207267 : -106839333/4916362 : 1)
**u= 122/153 ;  23698433*x^2 - 37332*y^2 + 47456402*x*z + 47396866*z^2
; C5a (90059/53246 : 23116669/319476 : 1)  C5b (1465679/1410907 : 484530701/8465442 : 1)
**u= 128/9 ;  65669*x^2 - 2304*y^2 + 196874*x*z + 131338*z^2
; C5a (-1893/934 : 42539/44832 : 1)  C5b (-131/354 : 142373/16992 : 1)
**u= 128/117 ;  13850573*x^2 - 29952*y^2 + 27766682*x*z + 27701146*z^2
; C5a (-11429/7349 : -8651797/352752 : 1)  C5b (-95791/72012 : -78592187/3456576 : 1)
**u= 128/169 ;  28915909*x^2 - 43264*y^2 + 57897354*x*z + 57831818*z^2
; C5a (-29083/10010 : -10522431/189280 : 1)  C5b (11721/481 : -65699787/100048 : 1)
**u= 128/173 ;  30301693*x^2 - 44288*y^2 + 60668922*x*z + 60603386*z^2
; C5a (5360813/5060512 : 4849559211/80968192 : 1)  C5b (129673/32019 : -23003457/170768 : 1)
**u= 130/9 ;  65153*x^2 - 2340*y^2 + 197906*x*z + 130306*z^2
; C5a (-3323/1596 : -5207/9576 : 1)  C5b (-4249/5428 : -266375/32568 : 1)
**u= 130/41 ;  1685953*x^2 - 10660*y^2 + 3439506*x*z + 3371906*z^2
; C5a (-25627/8559 : -474421/17118 : 1)  C5b (585183/184909 : -20050323/369818 : 1)
**u= 130/73 ;  5381377*x^2 - 18980*y^2 + 10830354*x*z + 10762754*z^2
; C5a (-931527/939728 : -31450429/1879456 : 1)  C5b (242899/5725344 : 93118993/3816896 : 1)
**u= 130/101 ;  10316713*x^2 - 26260*y^2 + 20701026*x*z + 20633426*z^2
; C5a (-1369729/684911 : -38268843/1369822 : 1)  C5b (-139406923/108266146 : 4490521823/216532292 : 1)
**u= 130/153 ;  23696417*x^2 - 39780*y^2 + 47460434*x*z + 47392834*z^2
; C5a (-802541/292229 : 86032453/1753374 : 1)  C5b (319283/206888 : 82829497/1241328 : 1)
**u= 130/157 ;  24952537*x^2 - 40820*y^2 + 49972674*x*z + 49905074*z^2
; C5a (-2181743/3069602 : 157867773/6139204 : 1)  C5b (-2406491/4115749 : -220667111/8231498 : 1)
**u= 136/49 ;  2413717*x^2 - 13328*y^2 + 4901418*x*z + 4827434*z^2
; C5a (-2209/775 : 608109/21700 : 1)  C5b (-373655/474673 : -186344219/13290844 : 1)
**u= 136/89 ;  8005477*x^2 - 24208*y^2 + 16084938*x*z + 16010954*z^2
; C5a (-957371/1308445 : 98231637/5233780 : 1)  C5b (-667299/472433 : -37465479/1889732 : 1)
**u= 136/193 ;  37714741*x^2 - 52496*y^2 + 75503466*x*z + 75429482*z^2
; C5a (-10859069/15476891 : -1730534553/61907564 : 1)  C5b (-115435/194841 : -7526737/259788 : 1)
**u= 144/113 ;  12914261*x^2 - 32544*y^2 + 25911466*x*z + 25828522*z^2
; C5a (5878998/2785591 : -2177018167/33427092 : 1)  C5b (-3219340/477893 : -666703963/5734716 : 1)
**u= 144/149 ;  22468877*x^2 - 42912*y^2 + 45020698*x*z + 44937754*z^2
; C5a (9164512/4601563 : -3987069257/55218756 : 1)  C5b (-296278/211211 : -62720023/2534532 : 1)
**u= 146/17 ;  271441*x^2 - 4964*y^2 + 628146*x*z + 542882*z^2
; C5a (288887/475012 : 13653489/950024 : 1)  C5b (-76975/64556 : -1197979/129112 : 1)
**u= 146/137 ;  18991681*x^2 - 40004*y^2 + 38068626*x*z + 37983362*z^2
; C5a (-4787277/5603036 : 246275377/11206072 : 1)  C5b (884863067/75090189 : -14000083699/50060126 : 1)
**u= 146/157 ;  24948121*x^2 - 45844*y^2 + 49981506*x*z + 49896242*z^2
; C5a (-2038229/2305611 : -9830003/419202 : 1)  C5b (624125/55374 : -10610313/36916 : 1)
**u= 146/173 ;  30296761*x^2 - 50516*y^2 + 60678786*x*z + 60593522*z^2
; C5a (-6099/3641 : -59/2 : 1)  C5b (-3566155/1765673 : -17681953/504478 : 1)
**u= 146/185 ;  34648609*x^2 - 54020*y^2 + 69382482*x*z + 69297218*z^2
; C5a (-1198751/1349876 : 68726637/2699752 : 1)  C5b (4111751/963017 : 261747043/1926034 : 1)
**u= 148/41 ;  1680949*x^2 - 12136*y^2 + 3449514*x*z + 3361898*z^2
; C5a (-7965976/3985913 : 129080559/7971826 : 1)  C5b (-213944/87887 : 3737281/175774 : 1)
**u= 148/53 ;  2823613*x^2 - 15688*y^2 + 5734842*x*z + 5647226*z^2
; C5a (301654/28491 : -8902079/56982 : 1)  C5b (-244506/149011 : -4869801/298022 : 1)
**u= 148/61 ;  3747469*x^2 - 18056*y^2 + 7582554*x*z + 7494938*z^2
; C5a (-5893234/9080987 : -275423919/18161974 : 1)  C5b (-6951940/3660741 : -6875489/348642 : 1)
**u= 148/137 ;  18991093*x^2 - 40552*y^2 + 38069802*x*z + 37982186*z^2
; C5a (-6738418/9476273 : 426272139/18952546 : 1)  C5b (-566366/29493071 : 1790087453/58986142 : 1)
**u= 148/153 ;  23691413*x^2 - 45288*y^2 + 47470442*x*z + 47382826*z^2
; C5a (-73008/139681 : 21223687/838086 : 1)  C5b (-47306/17497 : -4753837/104982 : 1)
**u= 160/61 ;  3743773*x^2 - 19520*y^2 + 7589946*x*z + 7487546*z^2
; C5a (-108247/294554 : -38475027/2356432 : 1)  C5b (-217491/451042 : 56946657/3608336 : 1)
**u= 160/89 ;  7998373*x^2 - 28480*y^2 + 16099146*x*z + 15996746*z^2
; C5a (487853/420939 : 134449663/3367512 : 1)  C5b (-1384787/1283084 : 174286691/10264672 : 1)
**u= 160/101 ;  10308013*x^2 - 32320*y^2 + 20718426*x*z + 20616026*z^2
; C5a (-565309/1033183 : 161675283/8265464 : 1)  C5b (-4757/12974 : 2202373/103792 : 1)
**u= 160/197 ;  39287917*x^2 - 63040*y^2 + 78678234*x*z + 78575834*z^2
; C5a (12769/63274 : 19758945/506192 : 1)  C5b (-161479/198738 : -4494455/176656 : 1)
**u= 162/13 ;  144953*x^2 - 4212*y^2 + 394882*x*z + 289906*z^2
; C5a (36411/26875 : 181057/11250 : 1)  C5b (-2529/863 : 266627/15534 : 1)
**u= 162/29 ;  825689*x^2 - 9396*y^2 + 1756354*x*z + 1651378*z^2
; C5a (-8237/7635 : 1200997/137430 : 1)  C5b (-140669/10646 : 22821029/191628 : 1)
**u= 162/41 ;  1676609*x^2 - 13284*y^2 + 3458194*x*z + 3353218*z^2
; C5a (-2207/4523 : -1015063/81414 : 1)  C5b (-29767/8523 : 4769291/153414 : 1)
**u= 162/89 ;  7997729*x^2 - 28836*y^2 + 16100434*x*z + 15995458*z^2
; C5a (1668899/1947220 : 1232731021/35049960 : 1)  C5b (-76355/15831 : 18852959/284958 : 1)
**u= 162/145 ;  21272081*x^2 - 46980*y^2 + 42649138*x*z + 42544162*z^2
; C5a (-472047/688891 : -276195995/12400038 : 1)  C5b (-5503959/7013644 : 2756353687/126245592 : 1)
**u= 164/85 ;  7292029*x^2 - 27880*y^2 + 14691642*x*z + 14584058*z^2
; C5a (158504/771 : -5152001/1542 : 1)  C5b (-3251176/6212971 : -224141425/12425942 : 1)
**u= 164/89 ;  7997077*x^2 - 29192*y^2 + 16101738*x*z + 15994154*z^2
; C5a (-463386/413551 : -13685881/827102 : 1)  C5b (-460124/24669 : 4830703/16446 : 1)
**u= 170/49 ;  2403313*x^2 - 16660*y^2 + 4922226*x*z + 4806626*z^2
; C5a (-49379/9818 : -6805689/137452 : 1)  C5b (-963/18046 : -4231137/252644 : 1)
**u= 170/81 ;  6617393*x^2 - 27540*y^2 + 13350386*x*z + 13234786*z^2
; C5a (169863/5179 : 48873683/93222 : 1)  C5b (38579/236642 : 14522201/608508 : 1)
**u= 170/109 ;  12006553*x^2 - 37060*y^2 + 24128706*x*z + 24013106*z^2
; C5a (-163381/19569 : 5220931/39138 : 1)  C5b (-368503/103447 : -10290377/206894 : 1)
**u= 170/113 ;  12906097*x^2 - 38420*y^2 + 25927794*x*z + 25812194*z^2
; C5a (-203483/801977 : -36654489/1603954 : 1)  C5b (-10653121/15967674 : 206639281/10645116 : 1)
**u= 170/117 ;  13838057*x^2 - 39780*y^2 + 27791714*x*z + 27676114*z^2
; C5a (-4973301/5361644 : 599253077/32169864 : 1)  C5b (-837877/419988 : -66716249/2519928 : 1)
**u= 170/137 ;  18984097*x^2 - 46580*y^2 + 38083794*x*z + 37968194*z^2
; C5a (202173/73513 : 1047863/13366 : 1)  C5b (-2625119/2783586 : 37688343/1855724 : 1)
**u= 170/157 ;  24940537*x^2 - 53380*y^2 + 49996674*x*z + 49881074*z^2
; C5a (1039019/403624 : -64789665/807248 : 1)  C5b (-15705421/41112277 : 2093074957/82224554 : 1)
**u= 178/29 ;  820249*x^2 - 10324*y^2 + 1767234*x*z + 1640498*z^2
; C5a (180589/111733 : -5667033/223466 : 1)  C5b (293513/175621 : -9131029/351242 : 1)
**u= 178/49 ;  2400529*x^2 - 17444*y^2 + 4927794*x*z + 4801058*z^2
; C5a (6589/21988 : 5939139/307832 : 1)  C5b (-966003/1068539 : -182674893/14959546 : 1)
**u= 178/61 ;  3737689*x^2 - 21716*y^2 + 7602114*x*z + 7475378*z^2
; C5a (4907127/4922302 : -289418191/9844604 : 1)  C5b (2132041/464182 : 69620071/928364 : 1)
**u= 178/81 ;  6614609*x^2 - 28836*y^2 + 13355954*x*z + 13229218*z^2
; C5a (18707/74952 : 32733289/1349136 : 1)  C5b (41469/11168 : -14717021/201024 : 1)
**u= 178/113 ;  12903313*x^2 - 40228*y^2 + 25933362*x*z + 25806626*z^2
; C5a (-110790157/87300304 : 3219374409/174600608 : 1)  C5b (-1007317/122172 : -10702819/81448 : 1)
**u= 178/125 ;  15796441*x^2 - 44500*y^2 + 31719618*x*z + 31592882*z^2
; C5a (-84521/217683 : -48023893/2176830 : 1)  C5b (-48821/7691 : -1581221/15382 : 1)
**u= 178/169 ;  28900609*x^2 - 60164*y^2 + 57927954*x*z + 57801218*z^2
; C5a (110107/160609 : -179443053/4175834 : 1)  C5b (-367075/270308 : -164217023/7028008 : 1)
**u= 180/49 ;  2399813*x^2 - 17640*y^2 + 4929226*x*z + 4799626*z^2
; C5a (7538/5579 : 7021727/234318 : 1)  C5b (46224/35267 : -43841851/1481214 : 1)
**u= 180/157 ;  24937037*x^2 - 56520*y^2 + 50003674*x*z + 49874074*z^2
; C5a (-28157762/19272823 : 2666244215/115636938 : 1)  C5b (-10312534/6018833 : 935737757/36112998 : 1)
**u= 194/13 ;  133561*x^2 - 5044*y^2 + 417666*x*z + 267122*z^2
; C5a (-26699/40725 : 258251/81450 : 1)  C5b (-9559/4441 : -112571/8882 : 1)
**u= 194/25 ;  595489*x^2 - 9700*y^2 + 1341522*x*z + 1190978*z^2
; C5a (-369565/539033 : 40631079/5390330 : 1)  C5b (-4902245/3255428 : -341769563/32554280 : 1)
**u= 194/29 ;  814297*x^2 - 11252*y^2 + 1779138*x*z + 1628594*z^2
; C5a (-226959/197590 : -3025127/395180 : 1)  C5b (14763519/91411 : 264033441/182822 : 1)
**u= 194/37 ;  1349161*x^2 - 14356*y^2 + 2848866*x*z + 2698322*z^2
; C5a (1145119/2769250 : 8516247/503500 : 1)  C5b (-197131075/386196838 : 8750751931/772393676 : 1)
**u= 194/41 ;  1665217*x^2 - 15908*y^2 + 3480978*x*z + 3330434*z^2
; C5a (-4797603/3235288 : 69405671/6470576 : 1)  C5b (267991629/884953316 : -30257251047/1769906632 : 1)
**u= 194/185 ;  34632289*x^2 - 71780*y^2 + 69415122*x*z + 69264578*z^2
; C5a (-1913747/3710572 : 180950307/7421144 : 1)  C5b (25573/15007 : -1901927/30014 : 1)
**u= 196/17 ;  254341*x^2 - 6664*y^2 + 662346*x*z + 508682*z^2
; C5a (484/5649 : 729689/79086 : 1)  C5b (3058/981 : -131113/4578 : 1)
**u= 196/97 ;  9492901*x^2 - 38024*y^2 + 19139466*x*z + 18985802*z^2
; C5a (-82662/153061 : -37133039/2142854 : 1)  C5b (-234986/252843 : 18904269/1179934 : 1)
**u= 196/109 ;  11997037*x^2 - 42728*y^2 + 24147738*x*z + 23994074*z^2
; C5a (-28678/112555 : 32894979/1575770 : 1)  C5b (-1950920/191521 : 416818261/2681294 : 1)
**u= 196/113 ;  12896581*x^2 - 44296*y^2 + 25946826*x*z + 25793162*z^2
; C5a (-52726/101385 : -26796239/1419390 : 1)  C5b (390212/3082027 : -1112004983/43148378 : 1)
**u= 196/121 ;  14792917*x^2 - 47432*y^2 + 29739498*x*z + 29585834*z^2
; C5a (-10278/55961 : -196353623/8617994 : 1)  C5b (16420/4549 : -58456997/700546 : 1)
**u= 196/149 ;  22451197*x^2 - 58408*y^2 + 45056058*x*z + 44902394*z^2
; C5a (-434512/61245 : -103758401/857430 : 1)  C5b (-214384/144169 : 44270909/2018366 : 1)
**u= 196/173 ;  30279661*x^2 - 67816*y^2 + 60712986*x*z + 60559322*z^2
; C5a (706856/470045 : -375096711/6580630 : 1)  C5b (1004232/1218919 : -750622893/17064866 : 1)
**u= 200/13 ;  131197*x^2 - 5200*y^2 + 422394*x*z + 262394*z^2
; C5a (-557/204 : 3341/816 : 1)  C5b (-1961/1592 : 57205/6368 : 1)
**u= 200/73 ;  5358277*x^2 - 29200*y^2 + 10876554*x*z + 10716554*z^2
; C5a (-89071/9706 : -21618483/194120 : 1)  C5b (-5071/6970 : -1991293/139400 : 1)
**u= 200/89 ;  7983973*x^2 - 35600*y^2 + 16127946*x*z + 15967946*z^2
; C5a (39/5542 : -471161/22168 : 1)  C5b (90767/22334 : 34649293/446680 : 1)
**u= 200/97 ;  9491317*x^2 - 38800*y^2 + 19142634*x*z + 18982634*z^2
; C5a (-183421/97131 : -40278239/1942620 : 1)  C5b (8581327/7452410 : -792074221/21292600 : 1)
**u= 200/109 ;  11995453*x^2 - 43600*y^2 + 24150906*x*z + 23990906*z^2
; C5a (-208939/13427 : 64979829/268540 : 1)  C5b (72279/6784 : 5279715/27136 : 1)
**u= 200/121 ;  14791333*x^2 - 48400*y^2 + 29742666*x*z + 29582666*z^2
; C5a (-49985/33827 : -143235537/7441940 : 1)  C5b (-1429/2002 : 8058833/440440 : 1)
**u= 200/169 ;  28892293*x^2 - 67600*y^2 + 57944586*x*z + 57784586*z^2
; C5a (2711/7127 : 65322483/1853020 : 1)  C5b (-10397/10634 : 11481395/552968 : 1)
**u= 200/181 ;  33146893*x^2 - 72400*y^2 + 66453786*x*z + 66293786*z^2
; C5a (881285/691199 : -735433539/13823980 : 1)  C5b (1366441/1092411 : 25605077/485516 : 1)
246

■これらのuについて、(3),(5a),(5b)を満たす有理数解(x,y,t)を持たないものもあれば、有理数解(x,y,t)を持つものもある。
これらのuを順に調べれば良い。

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 lt:= A, B, C, Dを満たすように、A,B,C,Cの符号を変更して、Dの小さい順に並び替えると、以下のようになる。

[参考文献]

Last Update: 2026.04.24
H.Nakao

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