# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - 3 x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.115026728081$, $\pm0.884973271919$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{7})$$ 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 principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2 4 74 256 1082 5476 16298 82944 261146 1170724

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 -1 9 15 33 83 129 319 513 1139

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
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.ac_f $4$ 2.16.ac_bh 2.2.a_d $4$ 2.16.ac_bh 2.2.c_f $4$ 2.16.ac_bh
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.ac_f $4$ 2.16.ac_bh 2.2.a_d $4$ 2.16.ac_bh 2.2.c_f $4$ 2.16.ac_bh 2.2.ab_ab $12$ (not in LMFDB) 2.2.b_ab $12$ (not in LMFDB)