curve #3713

y² + xy + y = x³ − x² − 1697652074924488760567x + 26922857125818100249183406500559

a-invariants: [1, -1, 1, -1697652074924488760567, 26922857125818100249183406500559]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 85197380776083810
discriminant (Δ): 4855332772164102056902166957704065685416609811392000000
Faltings height: 10.5315
naive height: 158.2642
primes of bad reduction: 2, 3, 5, 13, 37, 337, 373, 3253, 4813
regulator: 1903714.0615210610144222991754911724356348930535098
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:48:13 UTC
last updated: 2026-09-30 01:48:13 UTC

Witness: 6 independent points log in to add more points →

(32565940550403/1369, 1916690683463398/50653)
(148517173268707/6241, 1139866002758464834/493039)
(1247479754125947/52441, -30700259370662426/12008989)
(1451298031714203/61009, -125292958643562182/15069223)
(93034233729817587/3876961, 425463246363818304640894/7633736209)
(256740174301684083/10804369, 243275883642332641669778/35513960903)

Commentary

Specialization at t = 1068702/4849 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:48:13 UTC · history