# Properties

 Label 2.2.b_ab Base field $\F_{2}$ Dimension $2$ $p$-rank $2$ Ordinary Yes Supersingular No Simple Yes Geometrically simple No Primitive Yes 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: $\pm0.281693394748$, $\pm0.948360061415$ 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 $A(\F_{q^r})$ 7 7 196 259 1477 3136 14749 58275 268324 1109227

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 2 19 18 44 47 116 226 523 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.f 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.ab_ab $2$ 2.4.ad_f 2.2.ac_f $3$ 2.8.k_bp 2.2.a_d $6$ 2.64.as_ib
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.ab_ab $2$ 2.4.ad_f 2.2.ac_f $3$ 2.8.k_bp 2.2.a_d $6$ 2.64.as_ib 2.2.c_f $6$ 2.64.as_ib 2.2.a_ad $12$ (not in LMFDB)