curve #3166

y² + xy + y = x³ − 227204253761112793208000940016487352978x + 1318089867386926711140924601590234088488769981566633802648

a-invariants: [1, 0, 1, -227204253761112793208000940016487352978, 1318089867386926711140924601590234088488769981566633802648]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 2018326642496576098789458477481157728250403273123703770
discriminant (Δ): 96021171700353406555912759480309719179328745853239366947070046025851378212055522771921505906019063798681085437500
Faltings height: 20.7294
naive height: 276.5703
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 1406213, 1867993, 97776684006409817
regulator: 112665641451.65665734955217694907070392214324860101415199066495
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:26 UTC
last updated: 2026-09-26 20:40:26 UTC

Witness: 12 independent points log in to add more points →

(8145244832551235237, 2802136845278504101116818655)
(-2614487012653179451, -43522879055509397420848043025)
(7955909699716289909, 3748942862747705666913785655)
(-1305804858417159701, -40156540233256298925358896025)
(8120355012367564949, 2927152226216520118893211875)
(8806012412343667349, 442884651697222550228202375)
(8760832457062556699, 67105415368154448552857175)
(1723413480817630199, 30522807816313203931685157675)
(8209263705352116389, 2479721714605586058212981115)
(8221419028699601399, 2418353623765018460259967275)
(5015537671883797799, 17455867889605497871382497725)
(-4345785245986300288, -47152909934632349816986147545)

Commentary

Eroshkin (2006). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 12). Published independent points; exact point coordinates checked.

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