curve #5553

y² = x³ + x² − 57270457881880064056744x + 5215187312758600308803354062800836

a-invariants: [0, 1, 0, -57270457881880064056744, 5215187312758600308803354062800836]
c-invariants: [2748981978330243074723728, -4505921838239924558676079368708264640]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 81436544373824749184208
discriminant (Δ): 272254380799695876104541011098568688781908365670570076769588394230784
Faltings height: 11.9392
naive height: 168.8198
primes of bad reduction: 2, 3, 7, 19, 29, 71, 277, 317, 389, 449, 403957
regulator: 115034071.6193711234838073841728798398542450870025077
submitted by: Daniel Błażewicz
submitted at: 2026-10-23 02:09:24 UTC
last updated: 2026-10-23 02:09:24 UTC

(88100139422, 29213919217278180)
(-7347612617538150977/43811161, 29147379936696185358867290310/289986074659)
(3217669312462/9, 4710542063154107660/27)
(21698590104392/961, 1868439353592953632590/29791)
(146696106774420611/954529, 6148311630908012902490700/932574833)
(75753174210839686304/1309933249, 2170883305584335368997209442994/47410414081057)
(181846221953439475994/518700625, 1984719405689773077657962255472/11813406734375)

(u,v) = (155211,652703) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 7.

last edited by Daniel Błażewicz at 2026-10-23 02:09:24 UTC · history