Properties

Label 1.419.ak
Base Field $\F_{419}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{419}$
Dimension:  $1$
L-polynomial:  $1 - 10 x + 419 x^{2}$
Frobenius angles:  $\pm0.421453000851$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-394}) \)
Galois group:  $C_2$
Jacobians:  10

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 410 176300 73571630 30821471200 12914270735050 5411082293338700 2267243478329628670 949975016197928572800 398039531775790934530490 166778563814457086060307500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 410 176300 73571630 30821471200 12914270735050 5411082293338700 2267243478329628670 949975016197928572800 398039531775790934530490 166778563814457086060307500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{419}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-394}) \).
All geometric endomorphisms are defined over $\F_{419}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.419.k$2$(not in LMFDB)