curve #3096

y² = x³ + x² − 1773535921942955477718336x + 892419113191216158578012716419272964

a-invariants: [0, 1, 0, -1773535921942955477718336, 892419113191216158578012716419272964]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 635316132531616011586070488142160
discriminant (Δ): 12976150823148915240442687909395912145419897318937466440487393112627840000
Faltings height: 12.8187
naive height: 179.1187
primes of bad reduction: 2, 3, 5, 7, 13, 19, 31, 113, 233, 283, 823, 919, 1039, 1447, 832729
regulator: 510213377.3507153864329629559933110746638982321583884453353
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:06 UTC
last updated: 2026-09-26 20:37:06 UTC

Witness: 10 independent points log in to add more points →

(879730445262, 114155723149620048)
(888709182510, 134786983339820448)
(155666537634, 787471294356578700)
(246484957697976/289, 69149649256554891150/4913)
(116338786482, 829254308619428748)
(3414262679634, 5885389465456907700)
(110286793997136129/99856, 16723051335641584117067625/31554496)
(18166546926006879/21025, 217236971956647194216658/3048625)
(1860239690354633839689/24641296, 80220658918990503622110222057075/122319393344)
(97633026878740853250/4923961, 962583308829785569066230795228/10926269459)

Commentary

Dujella - Kulesz (2001). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 63). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:06 UTC · history