curve #6501

y² = x³ − 519763339618191410497x + 4558082975086992435076548407545

a-invariants: [0, 0, 0, -519763339618191410497, 4558082975086992435076548407545]
c-invariants: [24948640301673187703856, -3938183690475161463906137824118880]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 128819367123292381683043218556548306026945730239879506708
discriminant (Δ): 11346925133688086148169178863335000988077487702449546468867472
Faltings height: 10.6497
naive height: 154.7133
primes of bad reduction: 2, 19, 61, 218111, 333226853, 4089136407467, 93495040721692249378889723
regulator: 447892444475954009.6314067006165744903217618699997598753936319241052744540196
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:21 UTC
last updated: 2026-11-04 04:54:21 UTC

Witness: 19 independent points log in to add more points →

(493296602381, -346103618108677727)
(26049087967, -2948637055613003)
(256451464913/64, 815945909691627161/512)
(913206824497/81, 264064331623726637/729)
(11184625633003/961, 8700984934956826481/29791)
(40312238396835/6241, -597837007676058061415/493039)
(43920308379312/3721, -58999755325953665807/226981)
(5550889973728/441, -926574458315565599/9261)
(538990822117243/36481, 2246123019030883664161/6967871)
(7478262160554707/552049, -22434455521394824287313/410172407)
(8696914104523429/644809, -18840471982491066958229/517781627)
(17752093159954624/1697809, -1142240092176314643371731/2212245127)
(218422188937/16, 5226869351573083/64)
(335846010041/25, -833350609224161/125)
(118009111962264569/8596624, -2507227572421144377768589/25205301568)
(54185125593/4, 437353455090101/8)
(430074227429815/24649, -3463434686588676029545/3869893)
(14700794703, -306899063504741)
(34608911057449/2500, 15685173411079610707/125000)

Commentary

Mestre ten-root construction, roots 0,3,12,27,31,79,122,126,191,195: 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 >= 19: the 19 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 04:54:21 UTC · history