# Properties

 Label 2.4.ab_e Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $1 - x + 4 x^{2} - 4 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.278773483733$, $\pm0.627659203843$ Angle rank: $2$ (numerical) Number field: 4.0.2312.1 Galois group: $D_{4}$ Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+xy=x^5+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 416 3952 74048 1132816 16132064 262967728 4303225472 68606478928 1099812538656

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 24 64 288 1104 3936 16048 65664 261712 1048864

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.2312.1.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

 Subfield Primitive Model $\F_{2}$ 2.2.ab_a $\F_{2}$ 2.2.b_a

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.b_e $2$ 2.16.h_bo