curve #6496

y² + y = x³ + x² − 314114521949847734522930x + 193984355833118123854488928171215506

a-invariants: [0, 1, 1, -314114521949847734522930, 193984355833118123854488928171215506]
c-invariants: [15077497053592691257100656, -167602483439904523992599990087472801304]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 2201233101115632161881877774585510484195773539979034855
discriminant (Δ): -14272575936295091134683531097482395186459943673377980502198243939029296875
Faltings height: 12.7222
naive height: 176.0293
primes of bad reduction: 3, 5, 7, 47, 53, 242403607, 6898139839909, 5033058012181595839842686447
regulator: 14687330929661136.70455527840581323969178186767021282923860342995440916169851
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:46 UTC
last updated: 2026-11-04 04:53:46 UTC

Witness: 19 independent points log in to add more points →

(10125847323628, 32175217564641337126)
(252395464575685/4, -4009639754307699146779/8)
(1149565826424115/121, 38912850385741721587397/1331)
(3175323084954535/1089, -176304920923406263183156/35937)
(1412588029384540/841, -51188287505197298998882/24389)
(266442626685265/576, -5312127776444455358087/13824)
(364699793413873/841, -9103258389122922585586/24389)
(1356917825225365/25921, -1759147968122014443883078/4173281)
(-171028276297710/841, -12181157818974462331507/24389)
(398126462556881755/97969, 249172671705361758386041279/30664297)
(53014546420101460/6889, 12176729091365002397518019/571787)
(170545268101781185/157609, 66250778989465612054264126/62570773)
(7341481179054940/1681, 624572978530094656121777/68921)
(249342195018396910/4489, 124500703143321859679865931/300763)
(23152984955751310/24649, 3301187428950805984819366/3869893)
(228668966065, 366214551688905937)
(290736907494105145/222784, 148955229794900111822875021/105154048)
(967308028781231740/1787569, 1020894660841284044741552801/2389979753)
(-160861998878712335/929296, 441760691726189034578707153/895841344)

Commentary

Mestre ten-root construction, roots 0,20,22,99,116,144,145,161,174,195: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 19: the 19 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:53:46 UTC · history