curve #6299

y² = x³ + 33940198459x

a-invariants: [0, 0, 0, 33940198459, 0]
c-invariants: [-1629129526032, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 36861986285961791189792
discriminant (Δ): -2502206260276959037345316429861056
Faltings height: 5.0980
naive height: 84.3572
primes of bad reduction: 2, 29, 53, 59, 241, 1553
regulator: 212185982.082331737136995906887463724332021661969759826
submitted by: The Wren
submitted at: 2026-10-31 23:52:46 UTC
last updated: 2026-10-31 23:52:46 UTC

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

(76097/4, 204363929/8)
(2386961/16, 5859989255/64)
(2877317/49, 16071556654/343)
(2975625/24964, 7933421940675/3944312)
(587603951/12769, 58775886476890/1442897)
(1916900513/6724, 99925816977833/551368)
(9747984629/87025, 1852543300026742/25672375)
(46244339948/1329409, 53598427538481478/1532808577)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8. 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 23:52:47 UTC · history