curve #2939

y² = x³ − 398955873535051x + 2616220405824505272570

a-invariants: [0, 0, 0, -398955873535051, 2616220405824505272570]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 177546673115895824
discriminant (Δ): 1107136909990583002308401662040736074016292864
Faltings height: 7.3660
naive height: 112.4732
primes of bad reduction: 2, 13, 17, 53, 113, 1153, 1951, 3727
regulator: 3812099.177182706321544401632742249283566726791400953
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:16 UTC
last updated: 2026-09-26 20:29:16 UTC

Witness: 7 independent points log in to add more points →

(6915045, 13714414080)
(9351135301/225, -811561673671424/3375)
(161522698/9, -949531247522/27)
(6068183801871/167281, -12959532329150869410/68417929)
(16905639, -26519025390)
(73750760464105/3849444, -338196053479647340835/7552609128)
(381476071416541/6497401, -7057376513041968116640/16561875149)

Commentary

2-isogenous to curve#2085 via the rational 2-torsion point x=45357110. Independent witness points are the images of that curve's witnesses under the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:29:16 UTC · history