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curve #5669

y² = x³ − 2225627417x

a-invariants: [0, 0, 0, -2225627417, 0]
c-invariants: [106830116016, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 937925188033532192 ☆ record for quartic twists, rank ≥ 7
discriminant (Δ): 705565540590820631911150125632
Faltings height: 4.4169
naive height: 76.1835
primes of bad reduction: 2, 13, 29, 233, 1949
regulator: 129495.8253106252559134129717856602038002808200737817
submitted by: Michael Rubinstein
submitted at: 2026-10-24 04:20:13 UTC
last updated: 2026-10-24 04:20:13 UTC

Witness: 7 independent points log in to add more points →

(-6757, 3837976)
(-17541, 5800224)
(790569/16, 208973739/64)
(-13572, 5263674)
(6612957/49, 15932817732/343)
(-351364/49, 1354332538/343)
(-21853, 6180668)

Commentary

Rank 7, proved: 7 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 04:20:14 UTC · history

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