curve #68

y² = x³ + 46974552981863676115647417

a-invariants: [0, 0, 0, 0, 46974552981863676115647417]
rank (lower bound): ≥ 15
conductor (N): 238313731807359099208895797698912716157803778347364012
naive height: 131.7464
Faltings height: 9.0365
discriminant (Δ): -953254927229436396835583190795650864631215113389456048
primes of bad reduction: 2, 3, 1201, 6539317, 32344001, 61641367
regulator: 6004179571973955.9594170979361628807478033739008721205590284254100767
submitted by: David Renshaw
last updated: 2026-07-01 22:41:39

Witness: 15 independent points

(134903368, 7030622192107)
(192112402, 7352882361535)
(225027274, 7639981722671)
(225083527, 7640541103540)
(258845527, 8019816949840)
(323938954, 8998197347759)
(396959044, 10465465248949)
(493287703, 12923142427688)
(618875539, 16852542091394)
(800577328, 23666090147563)
(832883668, 24994838653493)
(900602302, 27882595405255)
(1938425152, 85618874611435)
(-3801659403949148854700590661/10563498456094496025, 20677120321600099128973780340089600900562/34332992588487460476311920125)
(-45422308525362180658274681/138657052360580625, 177509449917854185325666317221372981128/51631286351368162487765625)

Commentary

Provenance: Noam D. Elkies, "j = 0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields", NMBRTHRY posting, December 30, 2009. This is the Mordell curve y^2 = x^3 + 46974552981863676115647417. The submitted witnesses consist of integral points on this curve plus two points obtained from integral points on the 3-isogenous curve y^2 = x^3 - 27K via the dual 3-isogeny.

last edited by David Renshaw at 2026-05-29 11:30:59 · history