curve #3397

y² + xy + y = x³ − x² − 317164020078x + 68750282369702013

a-invariants: [1, -1, 1, -317164020078, 68750282369702013]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/7ℤ
conductor (N): 89409298
discriminant (Δ): -8202372212184655527515930624
Faltings height: 5.0256
naive height: 91.0617
primes of bad reduction: 2, 11, 43, 94513
regulator: 87.5764172035893289493707370263619222385856
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:34 UTC
last updated: 2026-09-26 21:20:34 UTC

Witness: 2 independent points log in to add more points →

(26338009/81, -106265557/729)
(24802251841/157609, 9429894986994991/62570773)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/7ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^3-t^2, c=t^2-t, at t = 1/44; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 89409298, torsion ℤ/7ℤ.

last edited by David Renshaw at 2026-09-26 21:20:34 UTC · history