Properties

Label 2.19.ak_ca
Base Field $\F_{19}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 52 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0969409553786$, $\pm0.438146995596$
Angle rank:  $2$ (numerical)
Number field:  4.0.38720.3
Galois group:  $D_{4}$
Jacobians:  10

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

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
$A(\F_{q^r})$ 214 131396 46981774 16884386000 6124262327854 2213732987579876 799087397110427254 288444712420098176000 104127327060645758306374 37589992633096691075945316

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 366 6850 129558 2473350 47054766 893962030 16983757278 322687625770 6131069385406

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.38720.3.
All geometric endomorphisms are defined over $\F_{19}$.

Base change

This is a primitive isogeny class.

Twists

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