curve #6512

y² + xy = x³ − x² − 1372770188470023264x + 639985038650044815789656644

a-invariants: [1, -1, 0, -1372770188470023264, 639985038650044815789656644]
c-invariants: [65892969046561116681, -552947073097120360132738315365]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 1271186737100143303970841727841690041815960146380218
discriminant (Δ): -11372036550097881997323150097271759114085579469517430228
Faltings height: 9.3327
naive height: 136.9701
primes of bad reduction: 2, 3, 7, 71, 2389, 550573638757, 36010374752173820440013854445407
regulator: 13990975023086335.952064615528307334890283619481792356849779003803748157909
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:40 UTC
last updated: 2026-11-04 04:55:40 UTC

Witness: 18 independent points log in to add more points →

(3620426666, -207666623831476)
(4941261594410, -10983907201793564278)
(-8675076124438/8281, 22961655059309733710/753571)
(-917389436179/2401, 3917435721035836691/117649)
(-60051783289/289, 148716242093321194/4913)
(42207262718/2209, -2572136049949255820/103823)
(6263766422078/22201, -54869764191914372882/3307949)
(301571383949/400, 46337349223387003/8000)
(1363644269777/961, 1172547344175199445/29791)
(3760539086, -220587175570900)
(8824615834/9, -415354207750162/27)
(1338207221, -34632931968616)
(13124013936/25, -1000074593931506/125)
(316087991447333/185761, -4550893275517452040163/80062991)
(495844142589581/1026169, -9833457189560558480489/1039509197)
(-4491426917582/7569, -23240725650877619650/658503)
(759418922, -5954139800950)
(-129067943605/289, -167617229168030729/4913)

Commentary

Mestre ten-root construction, roots 0,3,91,98,102,141,149,160,186,198: 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 >= 18: the 18 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 04:55:40 UTC · history