# Properties

 Label 2.4.a_af 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: $1 - 5 x^{2} + 16 x^{4}$ Frobenius angles: $\pm0.142549479296$, $\pm0.857450520704$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 144 4212 69696 1049052 17740944 268402692 4356000000 68719584492 1100510098704

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 7 65 271 1025 4327 16385 66463 262145 1049527

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{13})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-39})$$$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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.ad_h $3$ 2.64.a_el 2.4.d_h $3$ 2.64.a_el 2.4.a_f $4$ 2.256.o_vp