curve #6131

y² + xy + y = x³ − x² − 84901834894153305402x + 303937731692821368239156545401

a-invariants: [1, -1, 1, -84901834894153305402, 303937731692821368239156545401]
c-invariants: [4075288074919358659281, -262602200164258865821494141259929]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 6997011738463637227893675954536446387040433730
discriminant (Δ): -739375954467144903311737691117621487498952399011971072000000
Faltings height: 10.3033
naive height: 149.2964
primes of bad reduction: 2, 5, 29, 139, 6028103, 381358487, 75506740709120741075620403
regulator: 2604765951321901288.39070935738683406457185636274905302699058097415951314
submitted by: Michael Rubinstein
submitted at: 2026-10-29 12:56:04 UTC
last updated: 2026-10-29 12:56:04 UTC

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

(-10588186211, 125925604536319)
(-10363567521, 265974873256089)
(-10265761999, 306033740025799)
(-674735471471/289, 3437117879077232727/4913)
(-107451028759/121, 819006325472085989/1331)
(137793247701/25, 7313501602675159/125)
(229504973345/49, 32589823978222561/343)
(243062393449/49, 23870318646876057/343)
(580671339521/121, 111497127455076269/1331)
(10159495396129/729, 26607529057334518117/19683)
(13526478604389/289, 48849649441389129087/4913)
(27123119140181/8281, 186105687545905011529/753571)
(30961647126305/5041, 42606091621568006177/357911)
(82992885732649/11449, 322717030610090095557/1225043)
(112002462777241/32041, 1280762785628366537261/5735339)
(197364462431041/16641, 2109068996806894868311/2146689)
(1256865571385409/253009, 8783922005129675645073/127263527)

Commentary

Mestre ten-root construction, roots 0,1,9,15,16,28,32,67,102,114: 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 12:56:04 UTC · history