curve #3356

y² + xy = x³ − 4214703227356x + 3151874734423746320

a-invariants: [1, 0, 0, -4214703227356, 3151874734423746320]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4661767110
discriminant (Δ): 499979849606919636436410282650206310400
Faltings height: 6.1781
naive height: 98.8224 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 53, 101
regulator: 37.045958336204883939881531575268773036738518
submitted by: David Renshaw
submitted at: 2026-09-26 21:11:16 UTC
last updated: 2026-09-26 21:11:16 UTC

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

(7603808, 20262926396)
(-18928808/9, 44499213452/27)
(2438513828/1681, 20684439274696/68921)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/6ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=(10-2t)/(t^2-9), b=c+c^2, at t = -47/6; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 4661767110, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:11:16 UTC · history