curve #696

y² + xy + y = x³ − 95460745234882660501013x + 11363356629829709561964270673269092

a-invariants: [1, 0, 1, -95460745234882660501013, 11363356629829709561964270673269092]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 5211628064217511903251411089070580065454543197524287697413794
discriminant (Δ): -108119820848144924278449190787349743709472229539741564704598275496308
Faltings height: 11.9685
naive height: 170.3545
primes of bad reduction: 2, 3, 17, 31, 9161486801, 44366774625846872735099, 4054970795216337815466463
regulator: 1724165287499441277.0880959314429031110938689510853167432310635333627300286
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:21 UTC
last updated: 2026-09-10 16:08:21 UTC

Witness: 18 independent points log in to add more points →

(169808779698, -7051596339423014)
(181522844088, -4040749392924644)
(178017187587, 3329942588843536)
(-1317124148313/4, 673815232831029533/8)
(112212488010, -45435817670370974)
(-294518483391, 118031678663334472)
(1530357383035/9, 186440205533989972/27)
(8027298155517/16, -19173767442332518871/64)
(185770595501, -6374499517657260)
(4766981466723, -10386615207437980064)
(231680792301/16, -6394915681924624415/64)
(4287284661201/25, -751072045234975714/125)
(3632452225917/16, 2388249579650089699/64)
(35287948281045/121, -121461891720208800052/1331)
(3328308512745, -6046772366306681504)
(106349206286769/625, 106721195124025597222/15625)
(-3135772860131/9, -1302457888335596470/27)
(7150852652289, -19104594578073699722)

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 16:08:21 UTC · history