# Properties

 Label 1.193.ax Base Field $\F_{193}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{193}$ Dimension: $1$ L-polynomial: $1 - 23 x + 193 x^{2}$ Frobenius angles: $\pm0.189598946136$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 5

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 5 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 171 37107 7190208 1387542051 267786205371 51682553604864 9974730429197019 1925122952772657603 371548729890080996544 71708904872771704964307

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 171 37107 7190208 1387542051 267786205371 51682553604864 9974730429197019 1925122952772657603 371548729890080996544 71708904872771704964307

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{193}$.

## 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 1.193.x $2$ (not in LMFDB) 1.193.ac $3$ (not in LMFDB) 1.193.z $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.193.x $2$ (not in LMFDB) 1.193.ac $3$ (not in LMFDB) 1.193.z $3$ (not in LMFDB) 1.193.az $6$ (not in LMFDB) 1.193.c $6$ (not in LMFDB)