curve #6105

y² + xy + y = x³ − x² − 472418316145800992x + 120653394663409542397811395

a-invariants: [1, -1, 1, -472418316145800992, 120653394663409542397811395]
c-invariants: [22676079174998447601, -104244532887143488344216031305]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 131009313997870661224403129505597972969582
discriminant (Δ): 459039832713041460827865226177104599629400195365896192
Faltings height: 9.0637
naive height: 133.7035
primes of bad reduction: 2, 3, 7, 23, 31, 27357493, 146647847, 363488026492723498459
regulator: 873094578976225030.3099155916598027850235323704441574393098662481355487
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:29:25 UTC
last updated: 2026-10-29 02:29:25 UTC

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

(330335567, 802058935697)
(-253633214582183/1471369, 25048457484461470599293/1784770597)
(-52496376455/81, 9064530712451521/729)
(36910258681/81, 27933322763429/729)
(669054937981/400, 506591855955997219/8000)
(185749132297/361, 25478877413675603/6859)
(299196215089/625, 32039385976676513/15625)
(319026770669/25, 179937028409910893/125)
(1101316070677/121, 1152547348142701747/1331)
(2551181719591/46225, 96738712619829698791/9938375)
(2966259940405/2809, 4209004116403803085/148877)
(15433821254893/178929, 679267459060137483895/75686967)
(38927436085177/269361, 1040515059416357147243/139798359)
(39591044261749/120409, 38918574249770272291/41781923)
(225056259981253/137641, 3114366508373359799659/51064811)
(213141828789726535/207849889, 80100001563199815227017447/2996571849713)

Commentary

Mestre ten-root construction, roots 0,7,47,53,75,89,96,107,113,125: 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 2026-10-29 02:29:25 UTC · history