curve #6121

y² = x³ + x² − 60644523341955931815220x + 5753484063464912298436438516620224

a-invariants: [0, 1, 0, -60644523341955931815220, 5753484063464912298436438516620224]
c-invariants: [2910937120413884727130576, -4971010230851149848571566186722656960]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 12812290006126708208219509600625580071112139117656
discriminant (Δ): -26016874788140069728912815710527686485545853826622963559548468516608
Faltings height: 11.8528
naive height: 168.9934
primes of bad reduction: 2, 3, 7, 41, 103, 587087, 55772862889277, 551532355724953405051771
regulator: 1120114749907382356777.2168829263665795794898612643494017833201518380188989
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:14:07 UTC
last updated: 2026-10-29 03:14:07 UTC

Witness: 18 independent points log in to add more points →

(107443415702, 21862422841705614)
(-5965716966559/64, 52704901855722766671/512)
(2329046438369/16, 204064857834599097/64)
(4995690805616/25, 5022397094695402464/125)
(25077929631251/169, 10288704977791077096/2197)
(135973302581291/625, 835180363361526508536/15625)
(688722489807506/6241, 9922189865775253207974/493039)
(2866616322061940/22801, 37126617298997300495568/3442951)
(4041664140897376/27225, 21203426910054001422824/4492125)
(6655638529177456/26569, 343236446596044789270336/4330747)
(8431703193422516/10609, 740782603631545523574624/1092727)
(13939628213063714/22201, 1534583365180518932055378/3307949)
(35529268168582804/22201, 6621977305518633185597232/3307949)
(45266276751753029/376996, 3297246852909083227056897/231475544)
(185395739408581856/3481, 79826114326438531955933064/205379)
(293668626751092257/2298256, 33181317681957927552221991/3484156096)
(536016705898109156/4541161, 149855970683735259341498856/9677214091)
(253518154460556719846/1677312025, 435818959664229900342030627306/68694313983875)

Commentary

Mestre ten-root construction, roots 0,2,24,30,43,74,77,81,114,127: 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 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 03:14:07 UTC · history