curve #3087

y² = x³ + x² − 131460028018063043171643070840x + 8575269260274355355225951320318520536633088

a-invariants: [0, 1, 0, -131460028018063043171643070840, 8575269260274355355225951320318520536633088]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 145656839839920898983128703362734440
discriminant (Δ): 113631679952406301225296106120680091906448905169455844952725273966266776236538288131097600
Faltings height: 15.7833
naive height: 212.7591
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 47, 53, 149, 461, 7129, 100384235521
regulator: 98251214458.51571513529163459311540205513868911831414668388
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:40 UTC
last updated: 2026-09-26 20:36:40 UTC

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

(-14340159331129, 3233802420515284516080)
(34449745668128, 2021729252430622645152)
(-33000559371829, 3588535201500468410670)
(-409662858993467584/3721, -1057689894175075114716606480/226981)
(67478722455968, 108492602645051456832)
(33826207112344184/529, -7984776310091641866165360/12167)
(16527924784243888049794261/837870916, -67182221565453097205819286713168793495/24253011534536)
(-176874142617582664/841, 126542366209581341192475120/24389)
(34582091026953368/961, 58766179066002806311916352/29791)
(6264526202834822291/19321, -519646724481086706322242270/2685619)

Commentary

2-isogenous image of D-z4old9101112-023, rational kernel x=195804468832796. Source: https://dujella.github.io/tors/z4old9101112.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 20:36:41 UTC · history