# Properties

 Label 1.1024.acj 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: $1 - 61 x + 1024 x^{2}$ Frobenius angles: $\pm0.0978468837242$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-15})$$ 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 $A(\F_{q^r})$ 964 1046904 1073702236 1099510926000 1125899904573364 1152921505187041704 1180591620755126905036 1208925819616335707004000 1237940039285445752593241764 1267650600228231648146971404504

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 964 1046904 1073702236 1099510926000 1125899904573364 1152921505187041704 1180591620755126905036 1208925819616335707004000 1237940039285445752593241764 1267650600228231648146971404504

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-15})$$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

## Base change

This is a primitive isogeny class.

## Twists

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