curve #2868

y² + xy = x³ − 215903471220924151203065764861528595400x + 864328847120078351794179948910231841265079500777989160000

a-invariants: [1, 0, 0, -215903471220924151203065764861528595400, 864328847120078351794179948910231841265079500777989160000]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 3720844854479034579834388830
discriminant (Δ): 321376428424457778492877576634598596788604608494086573176396465756458283654487594306919350701858141280016000000000000
Faltings height: 21.0619
naive height: 276.4173
primes of bad reduction: 2, 3, 5, 11, 17, 19, 23, 43, 71, 97, 263, 337, 2063, 2957, 9479
regulator: 2397438.6293156798641730739724053253866523178340155
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:26:03 UTC
last updated: 2026-09-26 20:26:03 UTC

Witness: 6 independent points log in to add more points →

(797973359997676388302800/229441, 1370502119902560682800777417564179400/109902239)
(91083063688989370663487760/4566769, -654089009863248144840748763509810515720/9759185353)
(12736005904002286800, -13432631178313904311764275400)
(13528242951714550800, -20478852418660990325996675400)
(3098092920736259365399245450/18003049, -171826373366721687940399878058491459675300/76386936907)
(2628862212560550941711998344162300/83939513746201, -121160844676201841622349398429844901673414379857900/769041317955145378051)

Commentary

2-isogenous to curve#1903 via the rational 2-torsion point x=131121445710117831551/4. Independent witness points are the images of that curve's witnesses under 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 20:26:04 UTC · history