# Properties

 Label 2.2.ab_ab Base Field $\F_{2}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable No Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - x - x^{2} - 2 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.0516399385854$, $\pm0.718306605252$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Galois group: $C_2^2$ Jacobians: 0

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 is not principally polarizable, and therefore does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1 7 16 259 751 3136 18103 58275 258064 1109227

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 2 -1 18 22 47 142 226 503 1082

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
All geometric endomorphisms are defined over $\F_{2^{3}}$.

## 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.b_ab $2$ 2.4.ad_f 2.2.c_f $3$ 2.8.ak_bp 2.2.ac_f $6$ 2.64.as_ib
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.b_ab $2$ 2.4.ad_f 2.2.c_f $3$ 2.8.ak_bp 2.2.ac_f $6$ 2.64.as_ib 2.2.a_d $6$ 2.64.as_ib 2.2.a_ad $12$ (not in LMFDB)