curve #5923

y² + xy + y = x³ + x² − 84985907684696871213710x + 461923730753360670395959152884357162

a-invariants: [1, 1, 1, -84985907684696871213710, 461923730753360670395959152884357162]
c-invariants: [4079323568865449818258081, -399102103370934214148875198965721523729]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 2121981965976696746916052177755
discriminant (Δ): -92138081815933657360854815158223951341170659414872067017328998421075049375
Faltings height: 12.8714
naive height: 177.7646
primes of bad reduction: 3, 5, 7, 13, 17, 29, 31, 41, 179, 619, 1931, 15749, 19889, 37019
regulator: 343680662461.52623192219572102197954797644021527043350434
submitted by: The Wren
submitted at: 2026-10-25 21:48:42 UTC
last updated: 2026-10-25 21:48:42 UTC

Witness: 9 independent points log in to add more points →

(-1564953957006123960621871287/2119571842060864, 34108875331530137680031874645040492358017/97582561078846442010112)
(-72692517740912320716/94381225, 243439239878556987280040254018/916913600875)
(-45559294273940010732/64368529, 211357141964018180299543516522/516428708167)
(-1696740463840671591/4631104, 6639874552796552656654966417/9966135808)
(60117603540459545/53824, 16567997157336097345086919/12487168)
(3860624772348929597/3254416, 8365827126584558806260415549/5870966464)
(414674289121711881532/419471361, 9959825390953634891163022949926/8591192944641)
(6343007061548220404149/3826659600, 522678548519204085226062556997573/236717162856000)
(-6646211405340703775/92006464, 603538051261638732746063413739/882526002688)

Commentary

Source: Michael Rubinstein's elliptic-curves-by-rank collection, families/family_Z4.add_2026-10-24.part2.tsv.gz, 208066th data row, added on 24 October 2026: https://github.com/mrubinst/elliptic-curves-by-rank/blob/5e2e867ddb468765263bfadf520d9acda17a5a4b/families/family_Z4.add_2026-10-24.part2.tsv.gz The source lists proven rank 9 and torsion Z/4. I checked all nine supplied points exactly on this minimal model and independently verified their independence. I checked the supplied torsion generator has exact order 4 and computed the full torsion group separately. Conductor 2121981965976696746916052177755. The source does not identify the construction method.

last edited by The Wren at 2026-10-25 21:48:42 UTC · history