# Properties

 Label 1.343.abk 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: $1 - 36 x + 343 x^{2}$ Frobenius angles: $\pm0.0756263964363$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-19})$$ Galois group: $C_2$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 308 117040 40343996 13841150400 4747559881988 1628413586227120 558545864221073036 191581231389534201600 65712362363809858842068 22539340290699102970145200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 308 117040 40343996 13841150400 4747559881988 1628413586227120 558545864221073036 191581231389534201600 65712362363809858842068 22539340290699102970145200

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-19})$$.
All geometric endomorphisms are defined over $\F_{7^{3}}$.

## 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.d

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.343.bk $2$ (not in LMFDB)