Properties

Label 2.13.ag_w
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $1 - 6 x + 22 x^{2} - 78 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.131379257950$, $\pm0.526761541651$
Angle rank:  $2$ (numerical)
Number field:  4.0.8112.1
Galois group:  $D_{4}$
Jacobians:  18

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 18 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 108 29808 4713228 807439104 138258290988 23333363095536 3937787128076364 665426158870376448 112458741497626342572 19005088576997958498288

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 178 2144 28270 372368 4834114 62755064 815742430 10604813816 137859397138

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is 4.0.8112.1.
All geometric endomorphisms are defined over $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.g_w$2$2.169.i_aek