curve #4197

y² + xy + y = x³ + x² − 650585738720869344749x + 3173911523544766821670373312666

a-invariants: [1, 1, 1, -650585738720869344749, 3173911523544766821670373312666]
c-invariants: [31228115458601728547953, -2742259556576889399862715506253225]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1294818559182185565003
discriminant (Δ): 13271713351554838582136769683082996758130874765335979124779167009
Faltings height: 11.0001
naive height: 155.3868 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 7
primes of bad reduction: 3, 7, 17, 19, 23, 103, 137, 139, 149, 223, 347, 367
regulator: 574271.4382506085043501254883240251603007688762081773
submitted by: Manvir Jaswal
submitted at: 2026-10-12 08:26:28 UTC
last updated: 2026-10-12 08:26:28 UTC

Witness: 7 independent points log in to add more points →

(-26480400159, 1354014220930939)
(24489986028, 1388935407899026)
(242443568904, 118726480773418105)
(-80433792432507/6889, 1732272356985529145975/571787)
(122089063749/25, 42057012853612133/125)
(10042844118486/3025, 170531506633188342151/166375)
(4323804644773293/3964081, 12392920626800488070592995/7892485271)

Commentary

AI-assisted (Claude) search: Mestre-Nagao sieve over the universal Z/2xZ/4 family y^2 = x(x+a^2)(x+b^2) with (a,b) = (448324, 264607), then 2-descent (PARI ellrank, checked with mwrank). Rank exactly 7.

last edited by Manvir Jaswal at 2026-10-12 08:26:28 UTC · history