curve #3704

y² = x³ + x² − 10176238397624119849798012315398725175215x + 392750121982124018096479456962466953846105445381239468832938

a-invariants: [0, 1, 0, -10176238397624119849798012315398725175215, 392750121982124018096479456962466953846105445381239468832938]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/4ℤ
conductor (N): 17618290971556207682545290958840926789835323960 ☆ record for ℤ/4ℤ torsion, rank ≥ 12
discriminant (Δ): 806614322650607476631443999745256873912936495895224561963223330313077252622759798269982281086414472057895869300781250000
Faltings height: 21.8414
naive height: 287.9762
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 31, 43, 71, 73, 83, 227, 233, 359, 479, 1093, 1163, 1307, 1427, 4943, 71387
regulator: 3121650657432.8677052477540521728340191038904892355751750789374
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:46:31 UTC
last updated: 2026-09-30 01:46:31 UTC

Witness: 12 independent points log in to add more points →

(114716636197766052346, 857336867307199630618450837500)
(195364070310429571531, 2420983824233123693540174737545)
(-180686409811468638004931/3364, 172801161382077627803602932764296875/195112)
(-294677366174349387491/4, 6893989150101691940760420703125/8)
(104052664400770596961/16, 36589092709892255853958520115975/64)
(197155666653366737509/4, 835864754813879702252292984375/8)
(533729840577593012509/4, 9501395813659601910594788896875/8)
(10695217777023853962099/169, 103126133913615633929887502521875/2197)
(9707496827111768654716699/27889, 29094244319700487395770107368252585105/4657463)
(18322780137045138826521139/426409, 51956704448436928652587043883499921875/278445077)
(416356918383279285441195399/6692569, 366002276606927060207768605445470379025/17313676003)
(13818464431373299192706832826/112550881, 1190491932520025288326110663237984790510500/1194052296529)

Commentary

Specialization at t = 3549074/82573 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:46:31 UTC · history