curve #2164

y² + xy + y = x³ + x² − 7308940636813775218778413x + 7605385840556905645594196188764002531

a-invariants: [1, 1, 1, -7308940636813775218778413, 7605385840556905645594196188764002531]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 10598095089074167643269066006409167869829050
discriminant (Δ): 979702893705634841894394268859136703815446627633094192709560569856000000
Faltings height: 12.9277
naive height: 183.3670
primes of bad reduction: 2, 3, 5, 7, 13, 31, 61, 73, 163, 173, 193, 211, 229, 997, 1163, 1723, 2269, 6959, 21871
regulator: 169234069161.8668409633249045103498468998524231230172625988050490
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:31 UTC
last updated: 2026-09-26 14:17:31 UTC

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

(1595931177719, 75168645319103988)
(2197678472855, 1468675627887115572)
(2268120576215, 1641912624722503092)
(2613175574615, 2519998233000871092)
(3775602789655, 5816493024170785972)
(2483684904215, 2184801791379520692)
(14275501408435/9, 1444858895455190984/27)
(8029073059991, 21599155816257894708)
(5107111528215, 10172730488316800692)
(-12346788080985/4, 6959334482439168131/8)
(16443849788815/9, 15987145312702978364/27)
(566230925693335/361, 76520101334572115988/6859)
(-1015467311785, 3739018131668911092)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=1/17, t=-1/4; one of the smallest fibres of the family (|a|,b<=40 over 8 values of u), rank certified with PARI ellrank: 13 independent points.

last edited by Steps Unbounded at 2026-09-26 14:17:32 UTC · history