curve #3177

y² + xy = x³ − 35822192130572784206480514296239908919425x + 2609719568750620065454923921391767461604324824175741297455625

a-invariants: [1, 0, 0, -35822192130572784206480514296239908919425, 2609719568750620065454923921391767461604324824175741297455625]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/3ℤ
conductor (N): 5751679256865023413274743068599885347967822805404077230
discriminant (Δ): -236970537685396512783717319019850371091109730207165202086609487218990930925405052043298511702129909327649698787328000000
Faltings height: 21.9806
naive height: 291.7518
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 187172595299, 7081017707425445923
regulator: 310174806765.2842701719428032201676966654328826876068450587674922
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:00 UTC
last updated: 2026-09-26 20:41:00 UTC

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

(111765301087195517610, 46430435456204326675146597675)
(111116582336682124050, 34996683196066023464195962875)
(111913032824849252130, 49063192420997854527679793475)
(110123909838304503990, 18512376609554805482691445335)
(109146709187032115370, 10506129421757959855129586475)
(110382143345238886970, 22567063509763234684093270475)
(110245179394644403100, 20382223547410442956289336075)
(108522553923729999720, 17018033523451040060797113555)
(108512500742043656106, 17163334387154750309786061867)
(109741057847530716000, 13297340647117901361820923075)
(108902034932485796850, 12260441074498263650721717075)
(110291739877102075140, 21117334974942773428569817185)
(111294342929434080810, 38103895579546047175336101675)

Commentary

Eroshkin (2009). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 7). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:00 UTC · history