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curve #4039

y² = x³ + x² − 4975x − 101631

a-invariants: [0, 1, 0, -4975, -101631]
c-invariants: [238816, 86376320]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
conductor (N): 765253504
discriminant (Δ): 3564550821632
Faltings height: 1.1174 ☆ record for ℤ/2ℤ torsion, rank ≥ 5
naive height: 37.1503
primes of bad reduction: 2, 17, 137, 151
regulator: 140.534196088966821393400017992078412395158934616
submitted by: NDElkies
submitted at: 2026-10-06 19:07:42 UTC
last updated: 2026-10-06 19:07:42 UTC

Witness: 5 independent points log in to add more points →

(-23, 34)
(-40, 187)
(79, 68)
(1039, -33428)
(96, 561)

Commentary

2nd-smallest naïve height for this MW group, by exhaustive search

last edited by NDElkies at 2026-10-06 19:07:42 UTC · history

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