curve #4094

y² + xy + y = x³ + x² + 1671x + 254526

a-invariants: [1, 1, 1, 1671, 254526]
c-invariants: [-80207, -219309065]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/4ℤ
conductor (N): 1980381
discriminant (Δ): -28132205256831
Faltings height: 1.2603 ☆ record for ℤ/4ℤ torsion, rank ≥ 3
naive height: 38.4120 ☆ record for ℤ/4ℤ torsion, rank ≥ 3
primes of bad reduction: 3, 13, 17, 29, 103
regulator: 52.672594072313438181557931550109437557563802
submitted by: NDElkies
submitted at: 2026-10-09 14:34:43 UTC
last updated: 2026-10-09 14:34:43 UTC

Witness: 3 independent points log in to add more points →

(-30, 437)
(9, 515)
(201, 2859)

Commentary

Smallest naïve height of all curves with this MW group. This is [a,b,a*b,0,0] with (a,b)=(13,103). Curves 2-10 are at (a,b) = (24,481/4) [curve#1159], (33,111) [curve#4091], (39,129), (21,127), (3,211) [curve#4092], (19,-89) [curve#4093], (15,155), (11,181), (15,-145).

last edited by NDElkies at 2026-10-09 14:35:11 UTC · history