curve #1118

y² + xy = x³ − 233489241305x + 43447031653412025

a-invariants: [1, 0, 0, -233489241305, 43447031653412025]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/12ℤ
conductor (N): 36817770 ☆ record for ℤ/12ℤ torsion, rank ≥ 2
discriminant (Δ): -795295855006123967199000000000000 ☆ record for ℤ/12ℤ torsion, rank ≥ 2
Faltings height: 5.2598 ☆ record for ℤ/12ℤ torsion, rank ≥ 2
naive height: 90.1438 ☆ record for ℤ/12ℤ torsion, rank ≥ 2
primes of bad reduction: 2, 3, 5, 11, 31, 59, 61
regulator: 14.8260610854632474985206556795595187448168
submitted by: David Renshaw
submitted at: 2026-09-23 20:56:03 UTC
last updated: 2026-09-23 20:56:03 UTC

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

(282510, 5485995)
(135354715/484, 47972566885/10648)

Commentary

Rank exactly 2, torsion Z/12Z. Found by a small-parameter search of the universal family y^2=x^3+2(3t^8+24t^6+6t^4-1)x^2+(t^2-1)^6(1+3t^2)^2*x at t=1/11, followed by passage to the global minimal model. Family: Halbeisen, Hungerbuehler, Shamsi Zargar and Voznyy, Theorem 4.1, https://arxiv.org/abs/2106.06861 . PARI/GP ellrank gives lower and upper bounds [2,2]; two independent rational points are supplied. Conductor 36817770. This fills the rank-2 gap on the Z/12Z board.

last edited by David Renshaw at 2026-09-23 20:56:03 UTC · history