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curve #5695

y² = x³ + 440867857990043x

a-invariants: [0, 0, 0, 440867857990043, 0]
c-invariants: [-21161657183522064, 0]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 6219662982679319087228388539168
discriminant (Δ): -5484098993187586645132713663806424394919008448
Faltings height: 7.4660
naive height: 112.7729
primes of bad reduction: 2, 11, 17, 41, 61, 73, 157, 233, 353
regulator: 435957088.5039029740197211270410202618093952642870808004877
submitted by: Michael Rubinstein
submitted at: 2026-10-24 13:20:39 UTC
last updated: 2026-10-24 13:20:39 UTC

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

(11692175, 82177282770)
(25561327, 167243726562)
(56307893, 450946729434)
(89158149, 864893484066)
(280466527, 4710154854438)
(1261214284, 44796439546254)
(24994871/25, 2627310870294/125)
(487287337/36, 19852573396717/216)
(6678702913/784, 1451796403119201/21952)
(2357582128289/75076, 4354569712474069161/20570824)

Commentary

Rank 10, proved: 10 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 13:20:39 UTC · history

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