curve #3041

y² + xy = x³ − 74080202792053037257517366670840x + 245162596325324791545535916283786496196278857792

a-invariants: [1, 0, 0, -74080202792053037257517366670840, 245162596325324791545535916283786496196278857792]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 812040122999528003685943162590
discriminant (Δ): 53522188574629790367338571328926126150913249648354604215545060828773434442761454048709417369600
Faltings height: 17.0880
naive height: 231.7617
primes of bad reduction: 2, 3, 5, 7, 23, 37, 67, 97, 107, 113, 181, 353, 757, 797, 1500073
regulator: 27241955.4676389454962231578788778750436717627103715468
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:33 UTC
last updated: 2026-09-26 20:34:33 UTC

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

(4788825441078024, 15061463188506846389448)
(2502112947834762, 274718423698114510009170)
(3013816380468554, 221975741815687460563748)
(-2519466817783248, -644835137924363889045240)
(5855963060746032, 110300164969468654954440)
(9872268963343724, 689920984075167459213848)
(5101321879516304, 3132086030194574751848)
(-3629653943551056, -682810094828549484558072)

Commentary

Dujella - Peral (2012). Source: https://dujella.github.io/tors/z2z4old678.html (curve entry 6). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:34:33 UTC · history