curve #1384

y² = x³ − x² − 1358886867969640x + 19280709927026299248016

a-invariants: [0, -1, 0, -1358886867969640, 19280709927026299248016]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 134794654972773356297689104 ☆ record for ℤ/2ℤ torsion, rank ≥ 11
discriminant (Δ): 41067544627117908429577324461210976561152 ☆ record for ℤ/2ℤ torsion, rank ≥ 11
Faltings height: 7.2005
naive height: 116.1499
primes of bad reduction: 2, 3, 17, 29, 37, 41, 53, 73, 101, 109, 223, 13327, 29663
regulator: 17980304983.7402146072907109074406501765054753460153535786501
submitted by: jesper-petersen
submitted at: 2026-09-24 20:19:41 UTC
last updated: 2026-09-24 20:19:41 UTC

Witness: 11 independent points log in to add more points →

(21290386, -33206706)
(21297433, 104893488)
(21611938, -2635400718)
(21343261, 479894706)
(21293858, 72023538)
(21313946, 243025986)
(21429578, -1172242242)
(23428786, -17432387406)
(21321641, -305559204)
(21390941, 862519554)
(532539946/25, 17605466694/125)

Commentary

2-isogenous quotient of curve #1381, submitted by N. D. Elkies. Translating the rational 2-torsion point of #1381 to \((0,0)\) gives \(y^2=x^3-31915045x^2+254840875546408x\); quotienting by \((0,0)\) gives this curve. The 11 witness points are the explicit Vélu images of the 11 independent points on #1381. It has the same conductor and torsion \(Z/2Z\), while its minimal discriminant is \(41067544627117908429577324461210976561152\), smaller in absolute value than #1381 by a factor of approximately 20.07494.

last edited by jesper-petersen at 2026-09-24 20:19:41 UTC · history