curve #706

y² + xy = x³ + x² − 394473757008695118x + 86140454424768687268998288

a-invariants: [1, 1, 0, -394473757008695118, 86140454424768687268998288]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 7876458629296642392148676408365782668320176630
discriminant (Δ): 723051577062906525721823796018919068773910676869734100
Faltings height: 9.0690
naive height: 133.1626
primes of bad reduction: 2, 3, 5, 11, 19, 179, 809243, 8672243689441531494366410249103077
regulator: 106215503864915.5516064197527912012274495296088863723935301633388156405
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:38 UTC
last updated: 2026-09-10 16:08:38 UTC

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

(-217633671, 12715577147868)
(-474857286, 12899225291118)
(264727384, 514357141288)
(141545754, 5756687571768)
(31034034, -8598168322512)
(-140590689/4, -79984605459201/8)
(-713564271, -2071956422172)
(239227371, 2337069667098)
(82186542, 7367126607204)
(7243558389/16, 37425905074677/64)
(2319288024, -107922044136312)
(-386071479, 13449769580523)
(203414066, 3783503036446)
(914281542, -22130485138596)
(1664330688, -63560222427228)
(23683246464, 3643418656238868)

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 16:08:38 UTC · history