curve #4815

y² = x³ − 1849864165393131

a-invariants: [0, 0, 0, 0, -1849864165393131]
c-invariants: [0, 1598282638899665184]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 168987527427438277691643554724
discriminant (Δ): -1478302889935230053246497816725552
Faltings height: 5.0436
naive height: 83.8309
primes of bad reduction: 2, 3, 11777, 5817567089
regulator: 19795853848.69297509207398796361927247318402954320385841222
submitted by: The Wren
submitted at: 2026-10-20 21:49:01 UTC
last updated: 2026-10-20 21:49:01 UTC

Witness: 10 independent points log in to add more points →

(169635, 55059588)
(227635, 99727912)
(7454883, 20354492484)
(1640455/9, 1751018374/27)
(1831627/9, 2190044872/27)
(10317819/25, 32703271128/125)
(17695609/144, 4176392525/1728)
(71330065/484, 391392887399/10648)
(101621617/484, 916353777367/10648)
(49417435515/24649, 10984251096875916/3869893)

Commentary

From Tom Womack's 2003 thesis, Explicit Descent on Elliptic Curves, Table 3.3: https://johncremona.github.io/theses/womack.pdf. This is the rational 3-isogenous partner of the Mordell curve y^2=x^3+68513487607153, giving y^2=x^3-1849864165393131. I recovered ten rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [10,10]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:49:01 UTC · history