# 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

# Learn more

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 + 2 x^{2} )^{2}$ $1 + 4 x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.5$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 0

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.

## 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9$ $81$ $81$ $81$ $1089$

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

## 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.
TwistExtension degreeCommon 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.
TwistExtension degreeCommon 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)