curve #3175

y² + xy = x³ − 245159698188178088219881294961406816115x + 1510191009902655798002552220643158891490617937867360088417

a-invariants: [1, 0, 0, -245159698188178088219881294961406816115, 1510191009902655798002552220643158891490617937867360088417]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/3ℤ
conductor (N): 20766215158934528509379210237834484041686642848809798670
discriminant (Δ): -42218725295877017973742380027780444546936499991222418603452819965759646775156878393183006711999070568157716992000000
Faltings height: 20.9697
naive height: 276.8423
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 83, 89, 6099887, 1283387947, 585455194193
regulator: 479849819144.9245963779145914761202662830567966254099716613735536
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:56 UTC
last updated: 2026-09-26 20:40:56 UTC

Witness: 13 independent points log in to add more points →

(10833013559463040274, 11210358084599745780316885613)
(10706258660637427034, 10613283545662027955006055473)
(10395143048176178894, 9220066684034404984174781273)
(10137602529625035794, 8167669882323985109476036973)
(10038848157700788524, 7795499127400476443202567113)
(10824343292658261344, 11169050746067025317933688623)
(-4069189724279614606, -49400540986606610630851023667)
(-7414688387566494706, -54040095999701376974332025767)
(-976881606269713816, -41818067551922275505813314057)
(11184169214867230724, 12932997956751474213414498713)
(10145221804179288974, 8197181181207665106642796313)
(-8725386066273219586, -54635334476631791259074142967)
(13085600454673829294, -23298390072192192872367107527)

Commentary

Eroshkin (2008). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 6). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:56 UTC · history