curve #6104

y² + xy + y = x³ − x² − 4297086244037122052x + 3503885908179074526858758055

a-invariants: [1, -1, 1, -4297086244037122052, 3503885908179074526858758055]
c-invariants: [206260139713781858481, -3027357423738549762493948596585]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 112538881684078876199780374481021869696746
discriminant (Δ): -225646565432832671762104277315959534337598839140401020928
Faltings height: 9.5967
naive height: 140.3705
primes of bad reduction: 2, 3, 13, 23, 199, 146439585780487, 717542011250683003031
regulator: 2351837781524903699.814684507551567332458873742400892600828438234404139
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:29:18 UTC
last updated: 2026-10-29 02:29:18 UTC

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

(6655796045, 519373797555789)
(265305713589, 136648939801259805)
(-2645814312630739/3924361, 606911779276427574783441/7774159141)
(-24403843234411/10201, 9907475166618716865/1030301)
(-1265698184371/17161, 138953448383247577935/2248091)
(-16577132419/25, 9732135475169481/125)
(12333206659/9, 370881744091319/27)
(74325903697/81, 13312306040479345/729)
(369096220197/289, 48889677574690221/4913)
(650035877429/625, 197461103346538917/15625)
(9264923228597/5041, 15245799269102256315/357911)
(237340164570629/172225, 1008910650753825453987/71473375)
(3143194606962173/2699449, 39440188076385849213159/4435194707)
(3916232655242573/96721, 244761571291369818447387/30080231)
(8583843193242437/1560001, 745659033890418610636893/1948441249)
(1387773915783354916829/8106063823225, 1215367462756287623051087628603303/23078895902061245875)

Commentary

Mestre ten-root construction, roots 0,7,18,22,27,61,75,99,109,120: 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 02:29:18 UTC · history