# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 - 2 x^{2} )^{2}$ $1 - 4x^{2} + 4x^{4}$ 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $1$ $49$ $81$ $961$

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

## 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.
TwistExtension degreeCommon 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.
TwistExtension degreeCommon 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)