# Properties

 Label 1.343.ar Base Field $\F_{7^{3}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{7^{3}}$ Dimension: $1$ L-polynomial: $1 - 17 x + 343 x^{2}$ Frobenius angles: $\pm0.348223244152$ 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 $A(\F_{q^r})$ 327 118047 40366188 13841364891 4747558515717 1628413520361264 558545863791967707 191581231402213408083 65712362364002200523364 22539340290692787815555607

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 327 118047 40366188 13841364891 4747558515717 1628413520361264 558545863791967707 191581231402213408083 65712362364002200523364 22539340290692787815555607

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{7^{3}}$.

## 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 1.343.r $2$ (not in LMFDB) 1.343.au $3$ (not in LMFDB) 1.343.bl $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.343.r $2$ (not in LMFDB) 1.343.au $3$ (not in LMFDB) 1.343.bl $3$ (not in LMFDB) 1.343.abl $6$ (not in LMFDB) 1.343.u $6$ (not in LMFDB)