curve #3641

y² + xy + y = x³ − 9401745906324336774583921258x + 348880727126591652241550711849023093044968

a-invariants: [1, 0, 1, -9401745906324336774583921258, 348880727126591652241550711849023093044968]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 4858903795077400145101018027817292162914370
discriminant (Δ): 604927931528949936064851119231614985252709167711692665837745219425184361390826562500
Faltings height: 14.9118
naive height: 204.8457
primes of bad reduction: 2, 3, 5, 11, 19, 37, 67, 103, 229, 271, 631, 643, 1093, 1103, 1193, 1237, 1489, 2293, 19843
regulator: 395948805867751535.538982065881636893081713441714736536539309333691
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:25:13 UTC
last updated: 2026-09-29 19:25:13 UTC

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

(-6676247798144744711/63001, 6229025367045041504342558005/15813251)
(-72214492395752789/6724, 369299250156721340786162915/551368)
(-1167627466509864/49, 256536795112284919436665/343)
(25544700705016921/289, 2243001664415180277007315/4913)
(10808796856962554776/305809, 41675766241034160853367466935/169112377)
(26421245072990246569/1918225, 1251762649011620470492654132627/2656741625)
(77334535112489949289/3236401, 2161845307037460567778027721845/5822285399)
(312851945351432706699/6105841, 616680903014484491197103214455/15087533111)
(1246214398715912318662/19562929, 7964356572081712584179289201611/86526834967)
(43818955091918019033094/736525321, 213986702670965002895520971372420/19988560686619)
(510784333803417111889039/118243876, 364962174369734534205151606297221745/1285783907624)
(2900435326518811421392564/64262757001, 2091547048069415367054766669647757280/16290673162510501)
(7748059194553642502332054/20759334561, 20901273315800323948733455483662913805/2991025682883441)
(14556310661252676628478931679/38294673216361, 1703871622227946118508152135682376852586045/236977739129937069109)

Commentary

Specialization at T = 18/271 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 19:25:13 UTC · history