# Properties

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

# Related objects

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 - x + 2 x^{2} )^{2}$ $1 - 2 x + 5 x^{2} - 4 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.384973271919$, $\pm0.384973271919$ Angle rank: $1$ (numerical) Jacobians: 0

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

## 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $64$ $196$ $256$ $484$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $11$ $19$ $15$ $11$ $47$ $155$ $319$ $523$ $911$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
All geometric endomorphisms are defined over $\F_{2}$.

## 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_d$2$2.4.g_r
2.2.c_f$2$2.4.g_r
2.2.b_ab$3$2.8.k_bp
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_d$2$2.4.g_r
2.2.c_f$2$2.4.g_r
2.2.b_ab$3$2.8.k_bp
2.2.a_ad$4$2.16.ac_bh
2.2.ab_ab$6$2.64.as_ib