curve #695

y² + xy = x³ − 20028736733575489588360406091x + 893671825651003019065131087948865429489521

a-invariants: [1, 0, 0, -20028736733575489588360406091, 893671825651003019065131087948865429489521]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 95231609801426229953999389749708993933760516960154172648165810
discriminant (Δ): 169193642321060811256560695241468842453548289233199318853100698633549941158635582259200
Faltings height: 15.2632
naive height: 207.1145
primes of bad reduction: 2, 3, 5, 7, 13, 17, 29, 31, 1703173922378449, 1468108902457046219, 912835653667159934389
regulator: 165715651266355992.6900611506783568212620695852894375753339690692926539936801
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:19 UTC
last updated: 2026-09-10 16:08:19 UTC

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

(51243809076798, 43423653444506750337)
(-791187886914, 953686439576927184897)
(46876497237726, 240419400607510601217)
(114059845761108, -305083151704368920415)
(178676553984126, -1737616476727096849983)
(-55641447133662, -1354930020234105874455)
(216508372592766, 2589652788890564267457)
(39458734816254, 405955813928458488513)
(-31401464966274, 1221327172163114753217)
(51048970505598, -65260417477286016063)
(102422961539301/16, -56003528789491798878987/64)
(-110086216757634, -1328317511332991324223)
(761422521747966, 20666128983378971752257)
(-137177478028162, -1029467858093657723455)
(113575621519230, 289750614409021657281)
(-158573144872434, 286879385970244410657)
(-120168211517562, -1251084417208038063183)
(2447088254, 945316250554438109633)
(8898488109576, 846257024926559581257)

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 16:08:19 UTC · history