# Properties

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

# Related objects

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - 3 x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.115026728081$, $\pm0.884973271919$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{7})$$ Galois group: $C_2^2$ Jacobians: 0

This isogeny class is simple but not geometrically 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $4$ $74$ $256$ $1082$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $-1$ $9$ $15$ $33$ $83$ $129$ $319$ $513$ $1139$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
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.ac_f$4$2.16.ac_bh
2.2.a_d$4$2.16.ac_bh
2.2.c_f$4$2.16.ac_bh
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.ac_f$4$2.16.ac_bh
2.2.a_d$4$2.16.ac_bh
2.2.c_f$4$2.16.ac_bh
2.2.ab_ab$12$(not in LMFDB)
2.2.b_ab$12$(not in LMFDB)