curve #6310

y² = x³ − 3670496314042x

a-invariants: [0, 0, 0, -3670496314042, 0]
c-invariants: [176183823074016, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 862242764249338130328176896
discriminant (Δ): 3164858887986580780407697108581744773632
Faltings height: 6.2689
naive height: 98.4076
primes of bad reduction: 2, 61, 97, 241, 1061, 1213
regulator: 68229979.030080555830217403350449676418155288956369521331
submitted by: The Wren
submitted at: 2026-10-31 23:59:14 UTC
last updated: 2026-10-31 23:59:14 UTC

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

(-1893537, 401219451)
(-1286993, 1610028243)
(-838953, 1577619543)
(-175689, 799651737)
(-42777, 396149649)
(2289818, 1897724274)
(17832313/4, 67994609469/8)
(1729734361/36, 71882570840077/216)
(37840168186/7921, 6743004450240642/704969)

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 23:59:15 UTC · history