curve #2789

y² + xy = x³ − 186573780586500008328083787327652743900x + 980898400824382153587652612289503456380284993585235210000

a-invariants: [1, 0, 0, -186573780586500008328083787327652743900, 980898400824382153587652612289503456380284993585235210000]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 733110676848154824909690
discriminant (Δ): 9248148920983090344884484432626968249068095014341159769335996816895345665147250324161547357644810240000000000
Faltings height: 20.4684
naive height: 275.9793
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 41, 103, 157, 179, 1721, 5196749
regulator: 391794.8148281803252398215955119451296200526697
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:22:31 UTC
last updated: 2026-09-26 20:22:31 UTC

Witness: 4 independent points log in to add more points →

(2190443304866758281055800/175561, 1794821617365545295688542044700789900/73560059)
(184643250697384236577751695559905334614200/7155149337470205577489, 69933529551170720710364761418050399826466344301740149323759300/605240600845860567175438900043513)
(59574809831349313207687050/7447441, -11219328410770529568229883858556755850/20324066489)
(-2248465260103497317409480/190969, 3280715834147755189659157479396634260/83453453)

Commentary

2-isogenous image of D-z10-004, rational kernel x=-63094581573040382401/4. Source: https://dujella.github.io/tors/z10.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 20:22:31 UTC · history