curve #699

y² + xy = x³ + x² − 28531039064725089953x + 58454184659943741530872586853

a-invariants: [1, 1, 0, -28531039064725089953, 58454184659943741530872586853]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 5858355047047111277999276362771812224513127736824030
discriminant (Δ): 10292661149258010746542488627281052333491604383380874787600
Faltings height: 9.9882
naive height: 146.0062
primes of bad reduction: 2, 3, 5, 11, 271, 373, 55469, 90191, 1844287, 19034504786154264790192620749
regulator: 1607030376903663.80701034723347448891198396355935327254801102517189785432
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:26 UTC
last updated: 2026-09-10 16:08:26 UTC

Witness: 17 independent points log in to add more points →

(1723268774, -120021902063017)
(1605843114, -129534008094357)
(3252623194, 8046965579523)
(-4541198281, -307192311412402)
(-4987699171, -276906731280727)
(2833528331, -18994680164689)
(3679314019, -57340311705642)
(2559627418, 46852179182547)
(-3592231526, 338512755525683)
(17126886278, -2143286397973213)
(514862619, 209525637765798)
(6853478419, -429917774311177)
(14018519851/4, -310762957935491/8)
(2665090766, 36682990067531)
(46024244389/16, -871716500828353/64)
(48511288019, -10622555827877002)
(-1165740886, 300216834476843)

Commentary

Found by a search of rational specializations of Mestre families, using finite-field scores to select fibers and rational-point searches to construct the witness. Point independence checked with the site verifier.

last edited by David Renshaw at 2026-09-10 16:08:26 UTC · history