curve #2821

y² + xy = x³ − 15431325141995640971648011388750x + 28075861328873003447771017200972297203151562500

a-invariants: [1, 0, 0, -15431325141995640971648011388750, 28075861328873003447771017200972297203151562500]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/8ℤ
conductor (N): 552116366801381712611190
discriminant (Δ): -105351542462786455799105153105482610330131986512360815752533009850830722462115666323046400000000
Faltings height: 16.9502
naive height: 227.4256
primes of bad reduction: 2, 3, 5, 7, 13, 19, 79, 97, 991, 4451, 314908439
regulator: 650206.438105453436285000059984220164526811117872
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:23:55 UTC
last updated: 2026-09-26 20:23:55 UTC

Witness: 5 independent points log in to add more points →

(-3933616698398660, -167064625300798315877150)
(-304174496034500, -180946218395393828126750)
(30125238846195629500/1089, 163781914453907721439422841250/35937)
(4704373642915683100/841, 8334771827186805028417366250/24389)
(3873097779952529057020/3129361, 577240587069060000283078461520850/5535839609)

Commentary

Dujella - Peral (2012). Source: https://dujella.github.io/tors/z8old5.html (curve entry 7). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:23:56 UTC · history