curve #2977

y² + xy + y = x³ − 1317612112230610561758644823767449064x + 611501903663463837629781158845122941427734480390688926

a-invariants: [1, 0, 1, -1317612112230610561758644823767449064, 611501903663463837629781158845122941427734480390688926]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 220983838255926673069783016840911590
discriminant (Δ): -15139187414688141317262058211046790009890999899030865378231096782323429585890388166502847359599242093877917320
Faltings height: 19.7051
naive height: 261.2187
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 29, 37, 41, 43, 61, 103, 109, 211, 229, 499, 1429, 2640739
regulator: 3164174725.51702508233442425771810483567797408976789312
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:32 UTC
last updated: 2026-09-26 20:31:32 UTC

Witness: 8 independent points log in to add more points →

(-135953973075499104, -887763257997686062398308294)
(1312428102955619591424/1369, 24261673341929139546327450032593/50653)
(1002748957998087786, 546385665632257037405820511)
(2066335180868968446, 2590673627118633788956267747)
(610922047890120276, -185890969407468387448161134)
(7123288212271540966/9, 6854570113185352709625453241/27)
(30534920265309889337091519/54081316, 86728430328640305532629903178888443733/397713997864)
(-106023881338787565843970/208849, -102332980393027094218097439004952274/95443993)

Commentary

2-isogenous image of D-z6old678-010, rational kernel x=666411001163892393. Source: https://dujella.github.io/tors/z6old678.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 20:31:32 UTC · history