curve #3150

y² = x³ − x² − 50163588986898268948468352064401x + 56737927985190239949496934742877437522545251185

a-invariants: [0, -1, 0, -50163588986898268948468352064401, 56737927985190239949496934742877437522545251185]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 7396682463643816199120190345060611914560
discriminant (Δ): 6688088754991368442166725807900346051610163996607863773762636675725186051764035425731853189529600
Faltings height: 17.2729
naive height: 230.5921
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 31, 41, 43, 193, 229, 401, 457, 1031, 1193, 1949, 4703, 36389
regulator: 9430144518.43235965295905288812346438477621789910779887314988
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:39:42 UTC
last updated: 2026-09-26 20:39:42 UTC

Witness: 11 independent points log in to add more points →

(-484475637910219, 284477152446001008858672)
(822401256637789, 126647366250077572729312)
(-88753886282765, 247364998172561245002740)
(655741766959016, 155323966831302141321603)
(-5488330501674989, 408330803557522089307228)
(-212692404682652593/49, 150567701835014156375424720/343)
(7765644111643485, 368095491563641938713760)
(7319361658766048, 285819949391879004486615)
(6669922931624063, 137409361129262555031600)
(23004955806770853, 3328301870374901578219200)
(-3789712578770227, 438652434015468631229040)

Commentary

Dujella - Peral (2017). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 17). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:39:42 UTC · history