curve #3640

y² + xy = x³ − 39211499476584808467394120940x + 1333820595356270317012457647091674096534416

a-invariants: [1, 0, 0, -39211499476584808467394120940, 1333820595356270317012457647091674096534416]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 304327139592757069203804262254572708025894
discriminant (Δ): 3089954745649663912334772067123912757220807681183573537767821297718018893444133356634112
Faltings height: 15.4824
naive height: 209.1299
primes of bad reduction: 2, 3, 7, 11, 17, 19, 41, 53, 59, 97, 113, 157, 251, 271, 373, 431, 1471, 1523, 5569, 67751
regulator: 146996379335206.703767163164132602416230222333620897378767678458845
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:25:01 UTC
last updated: 2026-09-29 19:25:01 UTC

Witness: 14 independent points log in to add more points →

(-140148145316424, 2019039286251875890548)
(-88388380975496, 2027098167770009382580)
(-45657438921416, 1740384940487610288340)
(193670307863788, 1001967849743938185808)
(343197786067584, 5319772008421981751340)
(468438060110314, 9260507332124504125600)
(-59165253291925920440/310249, 236703185193410889964315457012/172808693)
(-3416617957242374696/16129, 751779883704445153311203884/2048383)
(-60909623840426774/2209, 160642428552230359217487200/103823)
(7323866022345208/9, 608732888150654179554908/27)
(49874703145357720/9, 11131270753803727089307612/27)
(627856720037590456/2209, 376493476293568849383245420/103823)
(2188654843421713240/97969, 20999435586510301866132657652/30664297)
(1434431601418880101912/48288601, 148269382800394110235857704372356/335557488349)

Commentary

Specialization at T = 853/2 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), 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 2026-09-29 19:25:01 UTC · history