# Properties

 Label 2.2.a_ab Base Field $\F_{2}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.209784688372$, $\pm0.790215311628$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=x^5+x^3+x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 16 76 576 964 5776 16636 57600 261364 929296

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 3 9 31 33 87 129 223 513 903

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.ad_f $3$ 2.8.a_l 2.2.d_f $3$ 2.8.a_l 2.2.a_b $4$ 2.16.o_dd
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.ad_f $3$ 2.8.a_l 2.2.d_f $3$ 2.8.a_l 2.2.a_b $4$ 2.16.o_dd 2.2.ad_f $6$ 2.64.w_jp