curve #6332

y² = x³ + 195843x

a-invariants: [0, 0, 0, 195843, 0]
c-invariants: [-9400464, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1227343380768
discriminant (Δ): -480733219439494848
Faltings height: 2.0823
naive height: 48.1688
primes of bad reduction: 2, 3, 97, 673
regulator: 1649.77359664740444792390701484271963684889826156
submitted by: The Wren
submitted at: 2026-11-01 01:57:35 UTC
last updated: 2026-11-01 01:57:35 UTC

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

(48, 3084)
(97, 4462)
(108, 4734)
(867, 28662)
(1164, 42486)

Commentary

OEIS A309028, a(5)=195843, contributed by Vaclav Kotesovec (14 July 2019); sequence by Seiichi Manyama. Curve y²=x³+195843x. PARI/GP 2.17.4 returned rank bounds [5,5]. All five displayed points pass exact substitution. Exact 2-descent quadratic characters at 12 good primes give matrix rank 6 with torsion-row rank 1, certifying five independent points modulo torsion. Torsion Z/2Z; conductor 1227343380768. Source: https://oeis.org/A309028

last edited by The Wren at 2026-11-01 01:57:35 UTC · history