curve #6495

y² + xy = x³ − 95856459473574582289205x + 10873501951626041519674551740024481

a-invariants: [1, 0, 0, -95856459473574582289205, 10873501951626041519674551740024481]
c-invariants: [4601110054731579949881841, -9394705686211801538080910073305974345]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 1527997307074255987966942096130361721461753649439001966
discriminant (Δ): 5292817332086209334734600129167610796935409470813874421081496062124032
Faltings height: 12.1347
naive height: 170.3650
primes of bad reduction: 2, 3, 7, 11, 2851, 25849, 21712742543531194873, 2066926182404924270286859
regulator: 58658407570421899.81579326650949625365950530991153560160103614838053916907909
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:39 UTC
last updated: 2026-11-04 04:53:39 UTC

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

(214053308795965/784, 1560686602366144318183/21952)
(17243810485708, -71594615598627890441)
(524884586388031/2500, 215203793040007798529/125000)
(61200528411586/289, -41396953396601660857/4913)
(51000923720398/361, -84277142998374485963/6859)
(-128463230094506/529, -1713368280550748322103/12167)
(2962040264527969/2304, 156887685081088639683097/110592)
(34384963312751206/22201, 6256885819402408028869379/3307949)
(54677556892710130/26569, 12647922812157497566527853/4330747)
(357612851920, 149425447114235599)
(4056660168229/16, 3427650814532782039/64)
(276690053023218994/543169, 122882621445724874702405303/400315553)
(136653106105506/289, 1311593768308346643975/4913)
(3092546287765218/10201, 101347492700677240178043/1030301)
(-47868402547482866/305809, 25107751712700516377287771/169112377)
(214484602726, -13447813478072177)
(324514801582887430/5004169, 786000067038592195003132043/11194326053)
(311225437318, 105764798077543339)
(302259061507162/289, 5045502038728841761079/4913)

Commentary

Mestre ten-root construction, roots 0,28,50,51,57,69,144,149,163,195: 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 04:53:39 UTC · history