curve #6187

y² + xy + y = x³ − 43334629206098088x + 3530877942247836371982322

a-invariants: [1, 0, 1, -43334629206098088, 3530877942247836371982322]
c-invariants: [2080062201892708201, -3050678545222223928231788725]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 65464962044891481098183123461289160971778
discriminant (Δ): -177615874612311788453316326764004537026481493604908
Faltings height: 8.4350
naive height: 126.5703
primes of bad reduction: 2, 3, 41, 266117731889802768691801314883289272243
regulator: 7640017432398204727.076683346374056257275234398494248702716219199834144
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:41:36 UTC
last updated: 2026-10-30 14:41:36 UTC

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

(-290385916385701/4618201, 24324673797296714854053/9924513949)
(97475104663/1156, 27126163503870965/39304)
(416663410274/3025, 69630216479870958/166375)
(3154297159991/16900, 3063570875730301221/2197000)
(851202342484/8649, 376621079973182185/804357)
(3722885494604/7225, 6671339518236104622/614125)
(15025928222825/135424, 14835864346606721331/49836032)
(5240285574539/1190281, 2373332888383203295377/1298596571)
(11386213189451/156025, 53622832875791126139/61629875)
(44350297717115/413449, 90810931720401361827/265847707)
(176366988042131/1646089, 724856779594131088059/2111932187)
(380371811281826/2430481, 2890542817985041846845/3789119879)
(3940940197927439/17656804, 165523403738292642297765/74193890408)
(99064472048900969/245862400, 27660961781393741827212843/3855122432000)
(160837774947784469/176810209, 62945773058288842762863465/2351045349073)
(4149866728519830917/27986947849, 2829912687612546448123857897/4682020466502757)

Commentary

Mestre ten-root construction, roots 0,5,8,26,33,45,67,74,115,125: 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 2026-10-30 14:41:36 UTC · history