# Properties

 Label 2.2.a_e Base field $\F_{2}$ Dimension $2$ $p$-rank $0$ Ordinary No Supersingular Yes Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 + 2 x^{2} )^{2}$ Frobenius angles: $\pm0.5$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

## 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 $A(\F_{q^r})$ 9 81 81 81 1089 6561 16641 50625 263169 1185921

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 13 9 1 33 97 129 193 513 1153

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.e 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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.a_ac $3$ 2.8.a_q 2.2.a_ae $4$ 2.16.aq_ds 2.2.ae_i $8$ 2.256.acm_chc
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.a_ac $3$ 2.8.a_q 2.2.a_ae $4$ 2.16.aq_ds 2.2.ae_i $8$ 2.256.acm_chc 2.2.ac_e $8$ 2.256.acm_chc 2.2.a_a $8$ 2.256.acm_chc 2.2.c_e $8$ 2.256.acm_chc 2.2.e_i $8$ 2.256.acm_chc 2.2.a_c $12$ (not in LMFDB) 2.2.ac_c $24$ (not in LMFDB) 2.2.c_c $24$ (not in LMFDB)