curve #4813

y² = x³ + 12502792885509

a-invariants: [0, 0, 0, 0, 12502792885509]
c-invariants: [0, -10802413053079776]
rank (lower bound): ≥ 9
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 7719497774712813154676004
discriminant (Δ): -67530166533187689477105682992
Faltings height: 4.2107
naive height: 73.8371
primes of bad reduction: 2, 3, 269, 1721436443
regulator: 52644998.735422853943234998090169710087240865055828115489
submitted by: The Wren
submitted at: 2026-10-20 21:46:30 UTC
last updated: 2026-10-20 21:46:30 UTC

Witness: 9 independent points log in to add more points →

(-20865, 1849122)
(-12225, 3267378)
(9043, 3638996)
(24223, 5168726)
(961503, 942820506)
(-92775/4, 1282851/8)
(-25383/4, 27996867/8)
(-6375/4, 28282851/8)
(18897/4, 28406457/8)

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^3-463066403167, giving y^2=x^3+12502792885509. I recovered nine rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [9,9]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:46:30 UTC · history