curve #6307

y² = x³ − 2715161751317x

a-invariants: [0, 0, 0, -2715161751317, 0]
c-invariants: [130327764063216, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 235907306746073553639503648
discriminant (Δ): 1281052992266291597026264928356580608832
Faltings height: 6.1935
naive height: 97.5032
primes of bad reduction: 2, 97, 101, 197, 829, 1697
regulator: 56176585.564549740277813725875828896866265817630235340333
submitted by: The Wren
submitted at: 2026-10-31 23:57:25 UTC
last updated: 2026-10-31 23:57:25 UTC

Witness: 9 independent points log in to add more points →

(-1481481, 878024406)
(-321652, 916547374)
(2312189, 2466475066)
(-277142161/169, 452850291074/2197)
(-202581072/361, 7960406342484/6859)
(-8121713/9, 35361938402/27)
(44292641/16, 236871266767/64)
(1599977461/729, 42274604474542/19683)
(27991358261/1369, 4667894886320882/50653)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by The Wren at 2026-10-31 23:57:26 UTC · history