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curve #4081

y² = x³ − 145775195951186x + 677444255404678004089

a-invariants: [0, 0, 0, -145775195951186, 677444255404678004089]
c-invariants: [6997209405656928, -585311836669641795532896]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/4ℤ
conductor (N): 765056790472
discriminant (Δ): 6997209405656912 ☆ record for ℤ/4ℤ torsion, rank ≥ 4
Faltings height: 6.0571
naive height: 109.4529
primes of bad reduction: 2, 13, 17, 269, 421, 3821
regulator: 76868.72183844591291983269526990459412835816668
submitted by: jesper-petersen
submitted at: 2026-10-09 05:39:38 UTC
last updated: 2026-10-09 05:39:38 UTC

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

(542612924857/77841, 857893191380/21717639)
(8573226466321/1229881, 1413841292688/1363938029)
(908307753294121/130302225, 11238785525456/1487399898375)
(17127700462305713664/2457071925025, 30261314101876837/3851472527836312625)

Commentary

Small-discriminant Z/4Z curve found by searching Tate normal form with a Nagao-type sieve.

last edited by jesper-petersen at 2026-10-09 05:39:38 UTC · history

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