# Properties

 Label 2.2.a_ae Base Field $\F_{2}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

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

This isogeny class is simple but not geometrically 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 $A(\F_{q^r})$ 1 1 49 81 961 2401 16129 50625 261121 923521

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 -3 9 1 33 33 129 193 513 897

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q(\sqrt{2})$$ ramified at both real infinite places.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.ae 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_c $3$ 2.8.a_aq 2.2.a_e $4$ 2.16.aq_ds 2.2.a_c $6$ 2.64.abg_ou
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.a_c $3$ 2.8.a_aq 2.2.a_e $4$ 2.16.aq_ds 2.2.a_c $6$ 2.64.abg_ou 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_ac $12$ (not in LMFDB) 2.2.ac_c $24$ (not in LMFDB) 2.2.c_c $24$ (not in LMFDB)