curve #704

y² = x³ + x² − 278772141172566912x + 56713426965241990382019604

a-invariants: [0, 1, 0, -278772141172566912, 56713426965241990382019604]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 1218797122339441835079463005318751008800638969328
discriminant (Δ): -2964114601529522542913254028935202453403153973405696
Faltings height: 8.7861
naive height: 132.1233
primes of bad reduction: 2, 19, 93889, 335173, 3284851, 15652866918389, 2477790187716079
regulator: 155141860112241.6127941825198589013025726661523640202253722443137304555
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:36 UTC
last updated: 2026-09-10 16:08:36 UTC

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

(313202988, 353772200842)
(224720254, -2327202617244)
(88918894, 5712123942636)
(386667723, 2594739836032)
(432422155, -4126054885512)
(280764460, 759216503178)
(479442063, 5767593844858)
(88360910, -5724574918028)
(80849678, 5890948720052)
(201454244, 2954551964086)
(398537868, 2985449921462)
(-21514770, -7918408084588)
(3707896894/9, 92713427557316/27)
(1625200260, -62419989814922)
(3863775934/9, 108532617068476/27)
(1231957525/4, 2111422398441/8)

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 16:08:36 UTC · history