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curve #3737

y² = x³ − x² − 1325894564x + 12210761491524

a-invariants: [0, -1, 0, -1325894564, 12210761491524]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 7654310664
discriminant (Δ): 84771041302180102273802221824
Faltings height: 4.2538
naive height: 74.6297 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 4
primes of bad reduction: 2, 3, 7, 11, 13, 19, 41, 409
regulator: 72.60851105207944910584537310526707989987161950
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 00:05:31 UTC
last updated: 2026-10-01 00:05:31 UTC

Witness: 4 independent points log in to add more points →

(-5998, 4466280)
(5454, 2267496)
(663944, 540196566)
(-128420/9, 143442026/27)

Commentary

(u, v) = (266, 143) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 4.

last edited by Daniel Błażewicz at 2026-10-01 04:19:16 UTC · history

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