# Properties

 Label 1.343.a Base Field $\F_{7^{3}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{7^{3}}$ Dimension: $1$ L-polynomial: $1 + 343 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 344 118336 40353608 13841051904 4747561509944 1628413678617664 558545864083284008 191581231352883840000 65712362363534280139544 22539340290701753210883136

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 344 118336 40353608 13841051904 4747561509944 1628413678617664 558545864083284008 191581231352883840000 65712362363534280139544 22539340290701753210883136

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{7^{3}}$
 The base change of $A$ to $\F_{7^{6}}$ is the simple isogeny class 1.117649.bak and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $7$ and $\infty$.
All geometric endomorphisms are defined over $\F_{7^{6}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{3}}$.

 Subfield Primitive Model $\F_{7}$ 1.7.a

## Twists

This isogeny class has no twists.