curve #6184

y² + y = x³ + x² − 71144170500499672x + 7306006044364626753567322

a-invariants: [0, 1, 1, -71144170500499672, 7306006044364626753567322]
c-invariants: [3414920184023984272, -6312389242820558619226072024]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 37096807484877365829693257492215827538671
discriminant (Δ): -13050930684715903178723276951695078173127520730651
Faltings height: 8.3980
naive height: 128.0246
primes of bad reduction: 3, 29, 43, 30491, 92119, 3530432767011153628475732039
regulator: 936845305762893512.5574581911267190010353783651708592457070676603062519
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:41:15 UTC
last updated: 2026-10-30 14:41:15 UTC

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

(-1199135483/4, 10406748011093/8)
(-1521295009099/6241, 1571885367707517304/493039)
(-140544546526/5929, 1368010106552453594/456533)
(11178678377/64, 233677936584107/512)
(11308630289/64, 256948558622951/512)
(900259355261/45796, 23835291086885471977/9800344)
(532623855140/3481, 10303575169909874/205379)
(5259741984869/31684, 1496839454753445907/5639752)
(1540633994543/9409, 197572591835526145/912673)
(4141694772893/7921, 7495905766672849079/704969)
(16210888004285/196249, 122730025286248510376/86938307)
(16814815291616/105625, 4160525777135549086/34328125)
(51623498764853/337561, 9982203872099362448/196122941)
(196113154223864/1408969, 529158045215591616818/1672446203)
(451958434211816/3031081, 601415263832578710781/5277112021)
(96331567495457645/598193764, 2328238721561318846199977/14630623079912)

Commentary

Mestre ten-root construction, roots 0,1,9,16,33,43,56,61,83,104: 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 14:41:15 UTC · history