# Properties

 Label 1.193.a Base Field $\F_{193}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{193}$ Dimension: $1$ L-polynomial: $1 + 193 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-193})$$ Galois group: $C_2$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

 $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 194 37636 7189058 1387413504 267785184194 51682554927364 9974730326005058 1925122950144000000 371548729913362368194 71708904873814507429636

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 194 37636 7189058 1387413504 267785184194 51682554927364 9974730326005058 1925122950144000000 371548729913362368194 71708904873814507429636

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-193})$$.
Endomorphism algebra over $\overline{\F}_{193}$
 The base change of $A$ to $\F_{193^{2}}$ is the simple isogeny class 1.37249.ow and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $193$ and $\infty$.
All geometric endomorphisms are defined over $\F_{193^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.