curve #6125

y² + xy + y = x³ − x² − 811621691337296202x + 277877419639547815275613065

a-invariants: [1, -1, 1, -811621691337296202, 277877419639547815275613065]
c-invariants: [38957841184190217681, -240086090393259027069273708825]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 20204071657297644439135497567378652495054663514
discriminant (Δ): 859668055556128483056597191597790994917899651413737472
Faltings height: 9.1527
naive height: 135.3270
primes of bad reduction: 2, 7, 53, 256496329586693513, 15165467407429906830935137
regulator: 2513625068873262007.82174940310712273154958292365100082344025551564138208
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:14:38 UTC
last updated: 2026-10-29 03:14:38 UTC

Witness: 17 independent points log in to add more points →

(-1038755439, 351782683127)
(408578865, 3804102867735)
(697603041, 7153404138071)
(9284494673, 890547386970487)
(-6556564485959/6889, 7844971374354549405/571787)
(-36013821951/289, 95403979917150631/4913)
(8332468999/25, 832805963277083/125)
(82219487113/121, 8430077923496117/1331)
(398890529265/6889, 8691834225493773765/571787)
(1702212475545/3721, 346067456890381875/226981)
(3239517185673/5041, 1666984150465109377/357911)
(4449981248227/9801, 1672352274092615495/970299)
(4628496385825/7569, 2106235747681769345/658503)
(5650458085519/9801, 1220147655445185245/970299)
(6973117140985/6889, 12700922523626252445/571787)
(28283283107129/42025, 52033599040002469507/8615125)
(601601387285385/76729, 14662284034448598968555/21253933)

Commentary

Mestre ten-root construction, roots 0,3,24,32,57,72,89,102,103,118: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 03:14:38 UTC · history