# Properties

 Label 1.313.an Base Field $\F_{313}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{313}$ Dimension: $1$ L-polynomial: $1 - 13 x + 313 x^{2}$ Frobenius angles: $\pm0.380247423634$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 7

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 7 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 301 98427 30674308 9597912051 3004147211821 940299071632704 294313622115692389 92120163576016007523 28833611193419038688164 9024920303509556734939707

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 301 98427 30674308 9597912051 3004147211821 940299071632704 294313622115692389 92120163576016007523 28833611193419038688164 9024920303509556734939707

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{313}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{313}$.

## 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 1.313.n $2$ (not in LMFDB) 1.313.aw $3$ (not in LMFDB) 1.313.bj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.313.n $2$ (not in LMFDB) 1.313.aw $3$ (not in LMFDB) 1.313.bj $3$ (not in LMFDB) 1.313.abj $6$ (not in LMFDB) 1.313.w $6$ (not in LMFDB)