# Properties

 Label 2.4.ad_j Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $1 - 3 x + 9 x^{2} - 12 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.272875599394$, $\pm0.469557725221$ Angle rank: $2$ (numerical) Number field: 4.0.3625.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2+(x^3+x+1)y=ax^5+ax^4+x^3+x+a+1$
• $y^2+(x^3+x+1)y=(a+1)x^5+(a+1)x^4+x^3+x+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 11 451 5456 65395 1046771 17000896 267003671 4242892995 68479461776 1102167168091

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 26 83 258 1022 4151 16298 64738 261227 1051106

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3625.1.
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.d_j $2$ 2.16.j_bp