# Properties

 Label 2.4.ab_ac 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: $1 - x - 2 x^{2} - 4 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.123737125646$, $\pm0.736024913457$ Angle rank: $2$ (numerical) Number field: 4.0.8405.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2+xy=ax^5+(a+1)x^3+ax^2+x$
• $y^2+xy=(a+1)x^5+ax^3+ax^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 200 3070 74000 1038550 17007800 276074830 4296884000 69074008390 1102005405000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 12 46 288 1014 4152 16846 65568 263494 1050952

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## 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 2.4.b_ac $2$ 2.16.af_bc