# Properties

 Label 1.64.ai Base Field $\F_{2^{6}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $1$ L-polynomial: $1 - 8 x + 64 x^{2}$ Frobenius angles: $\pm0.333333333333$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 57 4161 263169 16781313 1073709057 68718952449 4398044413953 281474993487873 18014398777917441 1152921505680588801

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 57 4161 263169 16781313 1073709057 68718952449 4398044413953 281474993487873 18014398777917441 1152921505680588801

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2^{6}}$
 The base change of $A$ to $\F_{2^{18}}$ is the simple isogeny class 1.262144.bnk and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{18}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.64.i $2$ (not in LMFDB) 1.64.q $3$ (not in LMFDB) 1.64.aq $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.64.i $2$ (not in LMFDB) 1.64.q $3$ (not in LMFDB) 1.64.aq $6$ (not in LMFDB) 1.64.a $12$ (not in LMFDB)