curve #6468

y² + xy + y = x³ − x² − 14701074255147819661142x + 668340710580824264426435539400805

a-invariants: [1, -1, 1, -14701074255147819661142, 668340710580824264426435539400805]
c-invariants: [705651564247095343734801, -577446373938656732425328376995489145]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 10883578061541468914591934184166771949462813702546719742
discriminant (Δ): 10376585390636390892311839439864587208459077840846239496217554747392
Faltings height: 11.6363
naive height: 164.7402
primes of bad reduction: 2, 3, 13, 109, 6113262985406879, 69800134932133388125350454276890833
regulator: 311780962404780454.8649661909433819425125693159743154142188947322967534526712
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:16 UTC
last updated: 2026-11-04 02:10:16 UTC

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

(8750257949329/9, -25691425140291627191/27)
(-9308892567, 28361682405949067)
(5023280920099/36, -7888601304755069825/216)
(86333421113173/1089, -38203885267757926141/35937)
(133858150257221/1681, 111625494779226529227/68921)
(148430729746533/1849, 186724771449804982089/79507)
(1456095287621/16, 587175247316376063/64)
(2586839256461/25, 1996227473963682519/125)
(1092757924693/9, 700614086616476869/27)
(-139502853847, 2076373110594699)
(-5723648407, 27428035661058699)
(84306542697, 5306611297883915)
(99987132713, 14072493453095499)
(100709274281, 14464931409351051)
(268573850537, 126857117233987467)
(274366825577, 131485252438894347)
(297281102261, 150234733551381927)
(2796088654073/49, 1339925927940728925/343)
(9695646975377/169, 8149471508755875303/2197)

Commentary

Mestre ten-root construction, roots 0,6,12,33,41,43,56,65,78,114: 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 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 02:10:16 UTC · history