# Properties

 Label 1.32.i Base Field $\F_{2^{5}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{5}}$ Dimension: $1$ L-polynomial: $1 + 8 x + 32 x^{2}$ Frobenius angles: $\pm0.750000000000$ 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 $A(\F_{q^r})$ 41 1025 32513 1050625 33546241 1073741825 34360000513 1099509530625 35184380477441 1125899906842625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 41 1025 32513 1050625 33546241 1073741825 34360000513 1099509530625 35184380477441 1125899906842625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{5}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{2^{5}}$
 The base change of $A$ to $\F_{2^{20}}$ is the simple isogeny class 1.1048576.dau and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{20}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{10}}$  The base change of $A$ to $\F_{2^{10}}$ is the simple isogeny class 1.1024.a and its endomorphism algebra is $$\Q(\sqrt{-1})$$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{5}}$.

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

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.32.ai $2$ 1.1024.a 1.32.a $8$ (not in LMFDB)