curve #6172

y² + xy = x³ − 227781760006021270350x + 1239644962466450800403457742596

a-invariants: [1, 0, 0, -227781760006021270350, 1239644962466450800403457742596]
c-invariants: [10933524480289020976801, -1071053247587413778269021021068145]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 921696670794521859371340115713061536866663911815042
discriminant (Δ): 92511493199181681140867470244016054332096813848356245845819392
Faltings height: 10.6375
naive height: 152.2384
primes of bad reduction: 2, 3, 7, 10463, 2097405985705915765239278809485264484980325027
regulator: 3751194039769578587125.7650982939562058037311580917546064443748586479716523
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:48 UTC
last updated: 2026-10-30 14:39:48 UTC

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

(-15841399770, 934153763484774)
(5360368764, 415536936027810)
(6384319084998890, 510119440091902353886544)
(-221494645289695420524/19141552609, 4039001423669049617394234987402/2648291228112977)
(-1363912545331659540/133795489, 2448130679338399862032971822/1547612421263)
(-5369639589865476/326041, 134765946799093800888486/186169411)
(-3749671525378884/237169, 108903600415519921152078/115501303)
(-3252252580728060/896809, 1206453736716114451852182/849278123)
(-340499430852724/1500625, 2088935024695739270783426/1838265625)
(-17397448923876/2809, 231255297531015154362/148877)
(50336223699048/3721, 180701954652924262698/226981)
(1361974277668620/128881, 5208975341000668470366/46268279)
(5163500908779066/769129, 76971896342134066073598/674526133)
(5826225655650876/1423249, 1040866797210302448655122/1697936057)
(16036567836331644/1452025, 466530002089710596871162/1749690125)
(53047264366535712/2271049, 10073435958680582219751222/3422470843)
(2020390374582514900/297459009, 392712921895892515486045078/5130275528223)
(22867048548802998204/221622769, 108235145670274904606263308174/3299298162103)

Commentary

Mestre ten-root construction, roots 0,8,18,22,27,35,49,65,71,103: 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 2026-10-30 14:39:48 UTC · history