curve #2965

y² + xy = x³ − 230893921241648413881171111278639894370x + 2045602011538050976181939572672311071842291227567176568900

a-invariants: [1, 0, 0, -230893921241648413881171111278639894370, 2045602011538050976181939572672311071842291227567176568900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 2023481314997324610295365836962841490
discriminant (Δ): -1019895925683278214944001291433644074444447196817683271647279897510218644995843142801831742809683783212112719071296000
Faltings height: 21.1510
naive height: 277.4492
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 47, 53, 61, 89, 103, 181, 233, 311, 491, 8252112961
regulator: 3585249174.44618690773625251153066683381659037966208019
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:40 UTC
last updated: 2026-09-26 20:30:40 UTC

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

(-704405305750855320019380/38809, 122535991127977320756609388044059790/7645373)
(-20616008069758170287567083098/8576797321, -40398181809257905679455705038411324854542344/794305776695131)
(-2711924026335295333380/529, -676834334416005785302653348284670/12167)
(10470341702715426252, 27855106510504734431583161262)
(41478515895255018604866912/410881, -264362407089879352329742020838604470098/263374721)
(-2658714628719404370899658652890/148148779801, -1153737046571190561739285169733375621566074470/57022613494184701)
(-10106706740325723389626812268/1933272961, -4740293897188215182430618975563531669944642/85004078822209)
(54181532059987767851127638073583917883939523064/5349278135513471920537643689, -10686356845797347697601337879351732565779993352931571486506134378967806/391239566844017838513203719330927198239637)

Commentary

2-isogenous image of D-z6old678-024, rational kernel x=9240887879951959140. Source: https://dujella.github.io/tors/z6old678.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 20:30:40 UTC · history