curve #5717

y² + y = x³ − 31302851647619

a-invariants: [0, 0, 1, 0, -31302851647619]
c-invariants: [0, 27045663823542600]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 16932128047594465965509583675
discriminant (Δ): -423303201189861649137739591875
Faltings height: 4.3637
naive height: 75.6726
primes of bad reduction: 3, 5, 5008456263619
regulator: 22388301032170.66403919915278481350962471821292648679306284
submitted by: jesper-petersen
submitted at: 2026-10-24 15:54:30 UTC
last updated: 2026-10-24 15:54:30 UTC

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

(35646997648001/693690244, 186675640193847449715/18270413646472)
(38268494849369/476636224, 229464118081208859869/10405922042368)
(3148825039492969/33969438864, 173187466998898464360341/6260839338146112)
(53045175644441/1047946384, -336501517671585417885/33924120342848)
(69349215973721/1900262464, -344567725232331344163/82836241330688)
(74483869361921/2129084164, -333342186036111349125/98240201495288)
(484823255599961/12089442304, 7658299924696040426787/1329258360209408)
(3298819331703449/45705308944, 181410388596425960001507/9771246588519872)
(7564968854180561/79863890404, 645746335607971430205037/22569695155951208)
(58668883543112081/313915919524, 14176465948573309389524587/175881439222745768)

Commentary

Constructed using the rational Bruin–Nasserden parametrization of the Burkhardt quartic with parameters (u,v,w)=(13/4,-10/4,4/4) and specialization t=-1151/1178. The model is the rational 3-isogenous partner of the resulting Mordell curve. A three-stage Mestre–Nagao sieve selected the specialization. Ten rational points were reconstructed through the 3-isogeny, transformed onto the minimal Weierstrass model, and checked by exact rational arithmetic. PARI/GP 2.17.4 returned matching computational rank bounds [10,10] for both isogenous curves.

last edited by jesper-petersen at 2026-10-24 15:54:30 UTC · history