curve #3472

y² + xy + y = x³ + x² − 904015764000240412335316946758212300240x + 7184469209853660442709629729284176740595203450045665168561

a-invariants: [1, 1, 1, -904015764000240412335316946758212300240, 7184469209853660442709629729284176740595203450045665168561]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 199500119263603519815431870142834901369753749343131572760786463422
discriminant (Δ): 24984952135622030580355191580126836742053515039282183032224538975281620665697360898253951150220631192853146758374490112
Faltings height: 21.4232
naive height: 280.7133
primes of bad reduction: 2, 3, 7, 17, 19, 29, 31, 37, 43, 47, 59, 61, 83, 163, 193, 263, 653, 701, 947, 1693, 2311, 4637, 10457, 56599, 57329, 7057319
regulator: 4512021069038085583348.61363632664949565605382890397516686488745090805396
submitted by: Steps Unbounded
submitted at: 2026-09-27 06:36:16 UTC
last updated: 2026-09-27 06:36:16 UTC

Witness: 17 independent points log in to add more points →

(-33424291830823600843, 7713352039302469893427633029)
(-22853316985022103359, 126129031273944385546553825237)
(-1292182814575415548842815/53824, 1531937636804193991090743695207993891/12487168)
(1161374041361208983051/25, 31976848787465082227223855165261/125)
(10548217603692010968261/169, 967457328321304210825416859656897/2197)
(11294050591060626731461/1369, 859545321481079463596302179207953/50653)
(2292186329875052117813293/22801, 3324347939567895002490140558737502219/3442951)
(6701930286236795218061013/1324801, 79827971691653499892626815283279672027/1524845951)
(10390451257052292064199101/81225, 32613606100305868570510369796403735769/23149125)
(251615090981840726613336613/237169, 3989622958614706494341266247571836039507/115501303)
(1162264485227951336938512597/5067001, 39293936457519141858606800904198788613159/11405819251)
(3804631167919789487641954093/1734489, 234654121795686378269890761356004949341649/2284322013)
(20079620248390600710896602285/13951625689, 126433081074179344781981973384163475234895761/1647924171507613)
(63381628989597542002095484077/7315551961, 982967124103581758585166267556160304159983/625706474776291)
(2463245100130886757513078631965/66951045001, 2668418806611354722336053921738843452116361273/17323515942963749)
(9343372809145410733570889197021/50014402321, 28203435734267064111243142558502207136850797923/11185170920666119)
(56292945441871788928842770203229/2264004724921, 30223171086319218389687054160711971586291635169/3406559613404356781)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=1546/487. Found by sweeping t=a/b with |a|,b<=8192 using the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 17 independent points, 2-Selmer upper bound 17.

last edited by Steps Unbounded at 2026-09-27 06:36:16 UTC · history