curve #6504

y² + xy + y = x³ − x² − 3706264993154900095730x + 87185257370033873299694608141297

a-invariants: [1, -1, 1, -3706264993154900095730, 87185257370033873299694608141297]
c-invariants: [177900719671435204595025, -75328062366908713292414683013403225]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 8228449224205764207124047078328908263712751841550
discriminant (Δ): -25460944043131256715924181305813687745322277173224203249745920000
Faltings height: 11.2100
naive height: 160.6143
primes of bad reduction: 2, 3, 5, 7, 53, 67, 744843943, 356371063635541, 395905015583754707
regulator: 1299989947977981.1644119293745432993604783712476006343664866194920276216312
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:43 UTC
last updated: 2026-11-04 04:54:43 UTC

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

(810839568025/9, 594333030616734757/27)
(3391339814369, -6244354911331460465)
(89680651201489/1521, 509024567759157594425/59319)
(33659191649, 754475085674255)
(8798207868181/289, 7874074289033508295/4913)
(-18819180119879/529, 160499781423411939385/12167)
(-39643194555599/729, 222617808826889319485/19683)
(-5192503161329/289, -59764358005205881775/4913)
(-149985374718959/7569, -8141145052376267619095/658503)
(25992055934844241/734449, 369590584321949965531975/629422793)
(-68257938215, -4705659831014329)
(-59875976815, -9717900796399057)
(1079963227835809/36481, 12867724998038845474225/6967871)
(371393895809, 223468678910881615)
(42118515169, 2408263132791695)
(32573900770849/961, 21110600480596036625/29791)
(-780530582161931/14641, -20450194275295186125925/1771561)
(35970298129, 640618415063855)

Commentary

Mestre ten-root construction, roots 0,15,39,67,68,92,108,170,174,195: 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 >= 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 04:54:43 UTC · history