curve #5672

y² = x³ + 33197384403x

a-invariants: [0, 0, 0, 33197384403, 0]
c-invariants: [-1593474451344, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3918458066490836147232 ☆ record for quartic twists, rank ≥ 8
discriminant (Δ): -2341486056605983575417867751604928
Faltings height: 5.0925
naive height: 84.2908
primes of bad reduction: 2, 3, 107, 241, 313, 457
regulator: 1282156.94896875965631914366497782495918128802365674341
submitted by: Michael Rubinstein
submitted at: 2026-10-24 04:21:02 UTC
last updated: 2026-10-24 04:21:02 UTC

(146697/4, 284738877/8)
(110137/4, -244614277/8)
(8667, 16981542)
(939, 5583294)
(75433, 54160894)
(106061931/289, 1219500082638/4913)
(2892, 9799542)
(3696217/4, 7243001227/8)

Rank 8, proved: 8 independent points listed, matching the 2-descent upper bound. Torsion Z/2. Member of the family y^2 = x^3 + a x.

last edited by Michael Rubinstein at 2026-10-24 04:21:02 UTC · history