# Properties

 Label 1.4.a Base Field $\F_{2^{2}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $1$ L-polynomial: $1 + 4 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$ Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 5 25 65 225 1025 4225 16385 65025 262145 1050625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 25 65 225 1025 4225 16385 65025 262145 1050625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

 Subfield Primitive Model $\F_{2}$ 1.2.ac $\F_{2}$ 1.2.c

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.4.ae $4$ 1.256.abg 1.4.e $4$ 1.256.abg 1.4.ac $12$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.4.ae $4$ 1.256.abg 1.4.e $4$ 1.256.abg 1.4.ac $12$ (not in LMFDB) 1.4.c $12$ (not in LMFDB)