curve #6071

y² = x³ − 23224755343755852x + 1454101348831682623642260

a-invariants: [0, 0, 0, -23224755343755852, 1454101348831682623642260]
c-invariants: [1114788256500280896, -1256343565390573786826912640]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 743330050898611727181276787476878581612392
discriminant (Δ): -111685685052660028964828249430830396158923766149888
Faltings height: 8.3497
naive height: 124.7960
primes of bad reduction: 2, 3, 7, 13, 169399, 669725691823818819025652417163929
regulator: 160249153910427164.5358090414635435076559612608410760214520201878728073
submitted by: Michael Rubinstein
submitted at: 2026-10-28 21:14:25 UTC
last updated: 2026-10-28 21:14:25 UTC

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

(-149463183, 1259544692133)
(-66012508862199/466489, 439974776065486823343/318611987)
(-6454524826814/97969, 50370264926068216402/30664297)
(-2213815514/25, 209773800197234/125)
(2017822641/16, 46634133885417/64)
(4107768846/25, 180021263406894/125)
(17677971114/25, 2300145800905638/125)
(26046635809/64, 3946409908252721/512)
(218753894706/2401, 36147178936197342/117649)
(247874727780/1681, 76613295128625450/68921)
(1985289026562/51529, 9184233373501968522/11697083)
(9239470065654/51529, 20443416718301696574/11697083)
(32545904826426/1681, 185665458472758058158/68921)
(35890365958548/237169, 136906426961342728326/115501303)
(57805332433306/36481, 437536165503560682338/6967871)
(104259853724316/546121, 804971618142813905238/403583419)

Commentary

Mestre ten-root construction, roots 0,7,44,49,64,66,82,101,124,139: 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 2026-10-28 21:14:26 UTC · history