curve #3053

y² = x³ + x² + 5173065494109668878164736x + 36560077211322832741553963860148538000

a-invariants: [0, 1, 0, 5173065494109668878164736, 36560077211322832741553963860148538000]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 1605314370710886136520856
discriminant (Δ): -586287953862184297173491668600362427662142067465377219320191579801443220149248
Faltings height: 13.6022
naive height: 186.5072
primes of bad reduction: 2, 3, 7, 13, 29, 31, 47, 151, 181, 313, 661, 1009, 3049
regulator: 204075651.023573976325145375822298771382258627071328879
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:06 UTC
last updated: 2026-09-26 20:35:06 UTC

Witness: 8 independent points log in to add more points →

(8005979673136/25, 773077626198404216316/125)
(447176857804821/4, 9458331276780524434839/8)
(109695168623004979/6241, 36755342162372551265315922/493039)
(-473838675552233600/4853209, 64197881107441293152439790620/10691619427)
(7235665272505600/6241, 3274758322973840973562740/493039)
(13250637035742211/5329, 3131441715915107961542526/389017)
(21848641838219959/841, 3245215452195114759382812/24389)
(1412364778696725525/2116, -1678505876527649130065425545/97336)

Commentary

2-isogenous image of D-z2z4old678-023, rational kernel x=1402149274312. 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 20:35:06 UTC · history