curve #6485

y² + xy + y = x³ − x² − 37094383587841840145515727x + 87430499935172500356082174687385737879

a-invariants: [1, -1, 1, -37094383587841840145515727, 87430499935172500356082174687385737879]
c-invariants: [1780530412216408326984754881, -75539951943981027920800025092429846130721]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 150042411546129109678533765048810322892047214841958925629770
discriminant (Δ): -35584006371595405919618132623445045676991719329491311922497843148435200000000
Faltings height: 13.5265
naive height: 188.2510
primes of bad reduction: 2, 3, 5, 19, 47, 109, 510233, 33567965926342972952914286179074343117294405193
regulator: 1231175078898561316.68919442333532976619392520030638069442882615653995497940727
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:27 UTC
last updated: 2026-11-04 04:52:27 UTC

Witness: 20 independent points log in to add more points →

(388789220075581/144, 4558710576205440021719/1728)
(250021739694887, -3952200724100531937694)
(4197673374126481/1600, 182927220506578725400109/64000)
(53757157894637743/3844, -11438966031880648474168487/238328)
(3560959092514907/961, -27568722824053110031454/29791)
(3008569351387, -1749718162766684194)
(7713853710742027/1521, 323539049453825871226514/59319)
(390790235470647763/20449, 233165601266030246424434842/2924207)
(762945802211798203/26569, 652504402556978185089136082/4330747)
(-338491189940237/49, -1284637531923217347142/343)
(160609276670917/49, 353041877375548232264/343)
(-3104882266103, 13140481611453225506)
(-6174692877613, -9003076788885515194)
(10876617205455103/5041, 1494640619514284735268058/357911)
(159986874441519217/48841, 11162826658999767782747716/10793861)
(1809850880664286507/591361, 729022025713742718472112654/454756609)
(2196019308547, -4069500111108609994)
(1651151333299891/529, 17451683112656620416874/12167)
(601145675252303/169, 1537726513690766161082/2197)
(88569344990989/25, 86576679031388595928/125)

Commentary

Mestre ten-root construction, roots 0,36,40,62,93,109,117,143,163,199: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 20: the 20 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:52:28 UTC · history