# Properties

 Label 2.5.ac_j Base Field $\F_{5}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $1 - 2 x + 9 x^{2} - 10 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.318486791133$, $\pm0.529524486268$ Angle rank: $2$ (numerical) Number field: 4.0.17984.1 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:

• $y^2=2x^6+x^5+3x^4+3x^2+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 23 1081 17756 381593 9826543 243683344 6044594327 152301779753 3823737714044 95455107487801

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 40 142 612 3144 15598 77368 389892 1957750 9774600

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.17984.1.
All geometric endomorphisms are defined over $\F_{5}$.

## 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 2.5.c_j $2$ 2.25.o_dn