curve #3714

y² + xy + y = x³ − x² − 106105666891528760567x + 420649558729925457528878500559

a-invariants: [1, -1, 1, -106105666891528760567, 420649558729925457528878500559]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 85197380776083810
discriminant (Δ): 12513634157431680639757513144354533613753995264000000000000
Faltings height: 10.1849
naive height: 149.9465
primes of bad reduction: 2, 3, 5, 13, 37, 337, 373, 3253, 4813
regulator: 76148.562460842440576891967019646897425395722140393
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:48:24 UTC
last updated: 2026-09-30 01:48:24 UTC

Witness: 6 independent points log in to add more points →

(237814467, 628831771092766)
(6041793597, 11232727163416)
(7113520323, 160695900034078)
(-288880348077/49, 314606274023391538/343)
(783703901383/49, 575477350569381238/343)
(34428032598843/5929, 8096670443160492278/456533)

Commentary

Specialization at t = -274449/134800 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:48:24 UTC · history