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curve #3410

y² + xy + y = x³ − 139631351475x + 20082708023947730

a-invariants: [1, 0, 1, -139631351475, 20082708023947730]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/3ℤ
conductor (N): 2082754778
discriminant (Δ): -4165509556 ☆ record for ℤ/3ℤ torsion, rank ≥ 4
Faltings height: 4.3662
naive height: 88.6004
primes of bad reduction: 2, 13, 31, 109, 151, 157
regulator: 5502.127405897948522788794551500421764218901272
submitted by: hdeping
submitted at: 2026-09-26 22:30:40 UTC
last updated: 2026-09-26 22:30:40 UTC

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

(3816224871/17689, -253780129574/2352637)
(154959, 46468198)
(9244253935/42849, -955011427910/8869743)
(311525795/1444, -5834461415/54872)

Commentary

Found by a Mestre-Nagao sieve on the Z/3 family E_(a,b): y^2+axy+b^3*y=x^3, t = -1609/1.

last edited by hdeping at 2026-09-27 08:28:37 UTC · history

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