curve #6185

y² + xy = x³ − x² − 114590917776624x + 496524101894893328804

a-invariants: [1, -1, 0, -114590917776624, 496524101894893328804]
c-invariants: [5500364053277961, -428996799285549596335845]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 62979444499453069527560665784219464167354
discriminant (Δ): -10202670008911397263464827857043553195111348
Faltings height: 7.0072
naive height: 108.8315
primes of bad reduction: 2, 3, 3498858027747392751531148099123303564853
regulator: 502583339240875792.7577453340423394722280642591724942385099514353545844
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:41:22 UTC
last updated: 2026-10-30 14:41:22 UTC

Witness: 16 independent points log in to add more points →

(8421358, 11342571820)
(122539766, 1351423465660)
(-4092698209024/466489, 9161501434369107026/318611987)
(-39487196339/46225, 242180394258828389/9938375)
(-17274995119/1600, 1395383627381259/64000)
(-747511322/361, 184681847063770/6859)
(-483568406/169, 62181820230460/2197)
(526689104/49, 7718934438510/343)
(12151708069/10000, 18948559485974047/1000000)
(28520371543/6724, 5135376981081895/551368)
(38614069105/5329, 2655524963692324/389017)
(161524335091/6724, 59392806101674189/551368)
(145480663894/1849, 54998582565021178/79507)
(588301282640/24649, 412370084234122854/3869893)
(6791946063046/1729225, 23521057077345490324/2273930875)
(8237269712866/1104601, 8736842763320149174/1160935651)

Commentary

Mestre ten-root construction, roots 0,4,64,78,83,88,107,116,132,138: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:41:22 UTC · history