curve #2161

y² = x³ − x² − 232512471305177343075x + 8568937089065100171604611728196

a-invariants: [0, -1, 0, -232512471305177343075, 8568937089065100171604611728196]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 16305275978428710457836900037023245758488
discriminant (Δ): -30915840540606445218989989935518049344204033623134155623607435568
Faltings height: 11.0553
naive height: 155.9745
primes of bad reduction: 2, 3, 7, 23, 29, 83, 109, 139, 149, 151, 179, 193, 337, 389, 1493, 2179, 3023, 115469
regulator: 792869416808.9781416847874616043870328190729208548903346221977955
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:21 UTC
last updated: 2026-09-26 14:17:21 UTC

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

(16249234573, 3013502396427597)
(20443767113, 3515669204543543)
(9341135709809/49, 28475798228503744217/343)
(-14007336155667/676, 37243524039386054337/17576)
(-420108929044/25, 347554794426211804/125)
(36794258273, 7058790390506897)
(-9833207324939/441, 15125563866075893369/9261)
(5186594277, 2739071478604725)
(1389781809, 2872016685288513)
(-24198398815, 160285889604689)
(-26109467354047/6889, 1752666947419516602035/571787)
(381609639136, 235567598473036134)
(44133778572156/961, 289985277247041964464/29791)

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=29/13, t=-1/7 (the 2-isogenous partner of that fibre); 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 14:17:21 UTC · history