# Properties

 Label 1.128.an Base Field $\F_{2^{7}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{7}}$ Dimension: $1$ L-polynomial: $1 - 13 x + 128 x^{2}$ Frobenius angles: $\pm0.305187096569$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$ Jacobians: 8

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 the Jacobians of 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 116 16472 2099948 268460656 34359708196 4398042893384 562949910253084 72057593939809248 9223372041104766164 1180591620785220370232

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 116 16472 2099948 268460656 34359708196 4398042893384 562949910253084 72057593939809248 9223372041104766164 1180591620785220370232

## Decomposition and endomorphism algebra

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

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{7}}$.

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

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.128.n $2$ (not in LMFDB)