Properties

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

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 51 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0769799514810$, $\pm0.443626041964$
Angle rank:  $2$ (numerical)
Number field:  4.0.46224.1
Galois group:  $D_{4}$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 213 130569 46774800 16859199849 6122283767853 2213551925280000 799063743815563173 288442310363975783049 104127192529352677789200 37589990180663150170659609

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 364 6820 129364 2472550 47050918 893935570 16983615844 322687208860 6131068985404

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.46224.1.
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_bz$2$(not in LMFDB)