curve #4862

y² = x³ + 136677874368461861091875625

a-invariants: [0, 0, 0, 0, 136677874368461861091875625]
c-invariants: [0, -118089683454351047983380540000]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/3ℤ
family: y² = x³ + k (Mordell curves)
conductor (N): 65605379696861693324100300
discriminant (Δ): -8070123459692610914690549350424656171231508362668750000
Faltings height: 9.2145
naive height: 133.8825
primes of bad reduction: 2, 3, 5, 11, 19, 23, 31, 83, 4201
regulator: 4280187.47996822128327285560703616355610776315172481174
submitted by: The Wren
submitted at: 2026-10-20 22:12:52 UTC
last updated: 2026-10-20 22:12:52 UTC

Witness: 8 independent points log in to add more points →

(248610979, 12330607336442)
(259337650, 12414503280025)
(823711075, 26373581199350)
(58174969086, 14031509431198959)
(-39439286271/400, 93198915410978817/8000)
(24612486921/64, 7123124452293531/512)
(44718594750/289, 58210804449583425/4913)
(1016602720650/529, 1034829899129894175/12167)

Commentary

From Elkies and Rogers (2004), Elliptic Curves x^3+y^3=k of High Rank: https://arxiv.org/abs/math/0403116. This is the rational 3-isogenous partner of the cube-sum curve with k=23381862574950. I recovered eight rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [8,8]. Rational torsion Z/3Z. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:12:52 UTC · history