curve #6106

y² + xy + y = x³ − x² − 692266474932009207x + 188382739401820381608488655

a-invariants: [1, -1, 1, -692266474932009207, 188382739401820381608488655]
c-invariants: [33228790796736441921, -162762686693643251124420209505]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 152829269309409686331640851550666966969954
discriminant (Δ): 5901598094299718325614404086495305511448419874263334912
Faltings height: 9.2318
naive height: 134.8498
primes of bad reduction: 2, 11, 13, 31, 17237679822852434731743836177607372769
regulator: 15633324452610939415.96172021830679064297761258593285682019230566327098
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:29:32 UTC
last updated: 2026-10-29 02:29:32 UTC

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

(-512217203, 20213739748522)
(307325235, 2158151671710)
(-677284929845/729, 107130404752103962/19683)
(9688729299/49, 2639775000742706/343)
(126974487019/5625, 5545165283374184122/421875)
(2181073429743/3481, 160616797939134378/205379)
(5662946311091/5329, 9938849992388824430/389017)
(90992767659119/42025, 810005677099750468262/8615125)
(125904878114627/36481, 1374396023455195081570/6967871)
(252816002820235/38809, 3988255825057070374950/7645373)
(1651083933394275/7187761, 124115356097295513700030/19270387241)
(11166548008615611/5929225, 1076965026468039360492586/14437662875)
(56708530601647935/4592449, 13474241984173546531937138/9841618207)
(175554716294162955/130256569, 61357376043046636795397030/1486618221997)
(240902438741723403/5368489, 118218802604233753831034870/12438789013)
(396698128161729619/1238769, 249855192001693803354647182/1378749897)

Commentary

Mestre ten-root construction, roots 0,13,29,52,54,55,69,71,106,125: 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 2026-10-29 02:29:32 UTC · history