curve #5419

y² + y = x³ − 2357095061516976785609184285626047

a-invariants: [0, 0, 1, 0, -2357095061516976785609184285626047]
c-invariants: [0, 2036530133150667942766335222780904392]
rank (lower bound): ≥ 9
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 9428380246067907142436737142504187
discriminant (Δ): -2400147559739975289389017909330207968162256197313345424663195438336163
Faltings height: 11.9917
naive height: 167.2086
primes of bad reduction: 3, 13, 23, 31, 43, 61, 73, 157, 199, 337
regulator: 14491371161687.368190809110772593034016550629214360173765
submitted by: The Wren
submitted at: 2026-10-21 22:36:25 UTC
last updated: 2026-10-21 22:36:25 UTC

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

(9165790760857/36, 25691656666726201967/216)
(2266728229583089/14400, -67884684612841020677687/1728000)
(5930433056062369/44100, 80091480203512336805653/9261000)
(465422286186947521/57600, -317519054363650281653812319/13824000)
(305179215546861457/1806336, -120542171581739587677113017/2427715584)
(578681535808377193/627264, 439548278760254029519448011/496793088)
(1543199831042552329/2722500, -1904603169274062219408348017/4492125000)
(91424327845739082697/557715456, 596039985484721722141704137467/13171008208896)
(798766994672348195257/5601624336, 9763829185288386618997938253043/419247971803584)

Commentary

Rational 3-isogeny partner of curve #3080: https://elliptic-rank.icarm.cloud/curve/3080 The source curve is attributed to Elkies–Rogers (2004) in Dujella’s Z/3Z table, curve entry 51: https://dujella.github.io/tors/z3old891011121314.html The nine source points were transported through the exact degree-3 isogeny and checked exactly on this minimal target model. Rank ≥9 is preserved under the rational isogeny. Rational torsion is trivial. This target is a cube-sum curve.

last edited by The Wren at 2026-10-21 22:36:25 UTC · history