# Properties

 Label 1.1024.acm Base Field $\F_{2^{10}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{10}}$ Dimension: $1$ L-polynomial: $( 1 - 32 x )^{2}$ Frobenius angles: $0$, $0$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial

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 a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 961 1046529 1073676289 1099509530625 1125899839733761 1152921502459363329 1180591620648691826689 1208925819612430151450625 1237940039285309906154946561 1267650600228227149696889520129

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 961 1046529 1073676289 1099509530625 1125899839733761 1152921502459363329 1180591620648691826689 1208925819612430151450625 1237940039285309906154946561 1267650600228227149696889520129

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 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 $\F_{2^{10}}$.

## 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 1.1024.cm $2$ (not in LMFDB) 1.1024.bg $3$ (not in LMFDB) 1.1024.a $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.1024.cm $2$ (not in LMFDB) 1.1024.bg $3$ (not in LMFDB) 1.1024.a $4$ (not in LMFDB) 1.1024.abg $6$ (not in LMFDB)