# Properties

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

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## Invariants

 Base field: $\F_{193}$ Dimension: $1$ L-polynomial: $1 - 14 x + 193 x^{2}$ Frobenius angles: $\pm0.331908682425$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$ Jacobians: 18

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 18 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 180 37440 7194420 1387526400 267784686900 51682526176320 9974730220761780 1925122954219545600 371548729951882295220 71708904873567206107200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 180 37440 7194420 1387526400 267784686900 51682526176320 9974730220761780 1925122954219545600 371548729951882295220 71708904873567206107200

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{193}$.

## 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 1.193.o $2$ (not in LMFDB) 1.193.ay $4$ (not in LMFDB) 1.193.y $4$ (not in LMFDB)