# Properties

 Label 1.1024.acf Base Field $\F_{2^{10}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{10}}$ Dimension: $1$ L-polynomial: $1 - 57 x + 1024 x^{2}$ Frobenius angles: $\pm0.150267280813$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 968 1047376 1073731736 1099512282528 1125899954494568 1152921506652542704 1180591620785220370232 1208925819616399499089728 1237940039285411746904576264 1267650600228229382588845215376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 968 1047376 1073731736 1099512282528 1125899954494568 1152921506652542704 1180591620785220370232 1208925819616399499089728 1237940039285411746904576264 1267650600228229382588845215376

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{10}}$.

 Subfield Primitive Model $\F_{2}$ 1.2.ab $\F_{2}$ 1.2.b

## Twists

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