curve #2909

y² + xy + y = x³ + x² + 31851784486271530053938080x − 6947650944050483344324943444909373055

a-invariants: [1, 1, 1, 31851784486271530053938080, -6947650944050483344324943444909373055]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 548088118837830139058490
discriminant (Δ): -2088998983067028655396166839158287669702702441390408514689431935582714215040000
Faltings height: 13.7176
naive height: 187.7830
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 47, 59, 61, 349, 21269
regulator: 1965606.217744365299818339264942494843646289636127023
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:27:56 UTC
last updated: 2026-09-26 20:27:56 UTC

Witness: 7 independent points log in to add more points →

(28681257415193, -156525477377836010207)
(12533925382477/9, 171002005391306491751/27)
(281675077042885, -4728351298329847751755)
(116874451549117/9, -1375839280045916825929/27)
(5201683925638293307/97969, -11930144300501355145229033239/30664297)
(7426292030485, -25281451881701963755)
(17058460537101553/7921, -5966702327085518757907283/704969)

Commentary

2-isogenous image of D-z2z4old678-067, rational kernel x=-108900008427. Source: https://dujella.github.io/tors/z2z4old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:27:56 UTC · history