curve #2748

y² + xy = x³ − 316709228995712421804107713926390256565x + 4102634832194430855328299974024166379582894591998979469217

a-invariants: [1, 0, 0, -316709228995712421804107713926390256565, 4102634832194430855328299974024166379582894591998979469217]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 244375491477491983033103910
discriminant (Δ): -5238140755504586562007518355137425452716584421467159549978367183198776223910511724358470002723210127655033041715200000
Faltings height: 21.2800
naive height: 278.8411
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 31, 37, 41, 79, 139, 101119, 6272219
regulator: 473614.1819781330615558096617411825239228104884
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:32 UTC
last updated: 2026-09-26 20:20:32 UTC

Witness: 4 independent points log in to add more points →

(6549660766270349842, 48054807623657732137411002775)
(118020126699863866269826/27321529, -9145690681381380222360495785537054976011/142809632083)
(103556388136794606782554/63001, -947013432357845045891383183425006547/15813251)
(64098390919985945531494308370890123631913139988088875673679728212967654143354570239814/1761925559294187155984837360322848519497467246435221195596960062121, 471989792300002435042372840062665712203520756307555034367990943910785778734006187171223099664131391183449040700319058201777110683/2338736714899638678778698812701839233678418950759085431813172729705797760171657198563577841458930581)

Commentary

2-isogenous image of D-z10-009, rational kernel x=178913345320035258735/4. Source: https://dujella.github.io/tors/z10.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:20:33 UTC · history