curve #73

y² + xy = x³ − 131092767138360259739530662694875901594863x + 11513825206543517171066572416002846205241167788788151682092217

a-invariants: [1, 0, 0, -131092767138360259739530662694875901594863, 11513825206543517171066572416002846205241167788788151682092217]
c-invariants: [6292452822641292467497471809354043276553425, -9947944978453598835810957246660421060029615177226994118242505625]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 288223154151653778337307627146202434973670195623654112440970678577409850
discriminant (Δ): 86914250236990244258622004202917499782878882540759655403422203387191788671749916349755506236622734346581449244863408384000000
Faltings height: 22.6751
naive height: 295.6438
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 41, 43, 47, 73, 89, 131, 139, 239, 271, 353, 421, 541, 1093, 1213, 1559, 1597, 2137, 41863, 11763979, 120637667
regulator: 79812493268609555646.384269644382536214993392374265076604666601157836610205
submitted by: David Renshaw
submitted at: 2026-06-01 14:26:54 UTC
last updated: 2026-07-01 22:33:41 UTC

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

(93922761628601831142, 172536183142846547019651383229)
(-50129317073713395258, -4237858386924835972521216331971)
(-85586222512760289858, -4701769957935207215670356805771)
(-112226086720247238258, -4981204691390384388312280789971)
(-6382114264687965138, -3514287164324372791040569133331)
(-61975801758421286388, -4404583464940048579421718953331)
(8805951679868879506902, 825658977348831750165246207555549)
(1063693026478103739858, 32796024716695803287365470950325)
(39058033423837004262, 2540311738090645849720634292669)
(306172033454233460742, 279024851751328167924122015229)
(-190899016575756710598, -5438977902059130788276281233951)
(2058431396010984516618, 91997445485354914650555923853105)
(-268567728817069704177/4, -35788685082941237533491020134023/8)
(350084920332270917759742, 207138045545955430308281145901442229)
(233038728147301046705862, 112497177953178712068114804246204669)
(523312432158494847942, 9285663592140230654731106188029)
(402009359485387238802, 4876746000213639259008862651449)
(5764113665374889509776198/11881, 10205816958752115947930462213486633025/1295029)

Commentary

Provenance: Maksym Voznyy (2023), as listed on Andrej Dujella’s high-rank elliptic-curve tables for torsion group Z/2Z. Dujella’s page lists this curve y^2 + xy = x^3 - 131092767138360259739530662694875901594863*x + 11513825206543517171066572416002846205241167788788151682092217 and the 18 independent points submitted here.

last edited by David Renshaw at 2026-06-01 14:27:16 UTC · history