# Properties

 Label 1.4.e 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: $( 1 + 2 x )^{2}$ Frobenius angles: $1$, $1$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial 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 $A(\F_{q^r})$ 9 9 81 225 1089 3969 16641 65025 263169 1046529

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 9 81 225 1089 3969 16641 65025 263169 1046529

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## 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.a

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.4.ae $2$ 1.16.ai 1.4.ac $3$ 1.64.q 1.4.a $4$ 1.256.abg
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.4.ae $2$ 1.16.ai 1.4.ac $3$ 1.64.q 1.4.a $4$ 1.256.abg 1.4.c $6$ (not in LMFDB)