curve #712

y² + xy + y = x³ − x² − 10078379364648458x + 390822864601184860438152

a-invariants: [1, -1, 1, -10078379364648458, 390822864601184860438152]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 46782040337234933341462490889078037190784315
discriminant (Δ): -468054313574035508081332221345225762093797071575
Faltings height: 8.0026
naive height: 122.1682
primes of bad reduction: 3, 5, 23, 29, 1117, 7919, 27767323, 185220684353, 34260508032133
regulator: 7347782734163.6964147664647382779511711156875128917677281272352801234
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:46 UTC
last updated: 2026-09-10 16:08:46 UTC

Witness: 15 independent points log in to add more points →

(13885118, 503540850513)
(-82002802, -815957578992)
(56545748, 41612565243)
(48926543, 121808551488)
(39292482, 235526427605)
(-114672462, -196458624312)
(668252138, 17089756964088)
(25174070, 391219691241)
(106777518, 729395588688)
(102809471, 664286046684)
(-62594262, 881180918648)
(-375348865/4, 5715049934199/8)
(-99922888, -632559875793)
(1491371786/25, -5441512219656/125)
(937519733/16, -2446281666843/64)

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 16:08:47 UTC · history