# Properties

 Label 1.16.ai Base Field $\F_{2^{4}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $1$ L-polynomial: $( 1 - 4 x )^{2}$ Frobenius angles: $0$, $0$ 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 $A(\F_{q^r})$ 9 225 3969 65025 1046529 16769025 268402689 4294836225 68718952449 1099509530625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 225 3969 65025 1046529 16769025 268402689 4294836225 68718952449 1099509530625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 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 $\F_{2^{4}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

 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 1.16.i $2$ 1.256.abg 1.16.e $3$ (not in LMFDB) 1.16.a $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.16.i $2$ 1.256.abg 1.16.e $3$ (not in LMFDB) 1.16.a $4$ (not in LMFDB) 1.16.ae $6$ (not in LMFDB)