# Properties

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

## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $1 - 11 x + 55 x^{2} - 143 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.129998747777$, $\pm0.292104599859$ Angle rank: $2$ (numerical) Number field: 4.0.6725.1 Galois group: $D_{4}$ Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2=2x^6+10x^4+12x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 71 26909 4949339 824733941 138120708816 23299305353501 3937346592497291 665439985622292389 112458147667396136951 19005123190067808033024

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 159 2253 28875 371998 4827063 62748045 815759379 10604757819 137859648214

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is 4.0.6725.1.
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.
 Twist Extension Degree Common base change 2.13.l_cd $2$ 2.169.al_ij