# Properties

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

# Related objects

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 + x - x^{2} + 2 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.281693394748$, $\pm0.948360061415$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Galois group: $C_2^2$ Jacobians: 0

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $7$ $196$ $259$ $1477$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $2$ $19$ $18$ $44$ $47$ $116$ $226$ $523$ $1082$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{3}}$ is 1.8.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
All geometric endomorphisms are defined over $\F_{2^{3}}$.

## 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.ab_ab$2$2.4.ad_f
2.2.ac_f$3$2.8.k_bp
2.2.a_d$6$2.64.as_ib
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.ab_ab$2$2.4.ad_f
2.2.ac_f$3$2.8.k_bp
2.2.a_d$6$2.64.as_ib
2.2.c_f$6$2.64.as_ib
2.2.a_ad$12$(not in LMFDB)