# Properties

 Label 1.27.ah Base Field $\F_{3^{3}}$ Dimension $1$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{3^{3}}$ Dimension: $1$ Weil polynomial: $1 - 7 x + 27 x^{2}$ Frobenius angles: $\pm0.264757707515$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-59})$$ Galois group: $C_2$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21 735 19908 532875 14352891 387409680 10460169993 282428545875 7625595497436 205891144928175

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 21 735 19908 532875 14352891 387409680 10460169993 282428545875 7625595497436 205891144928175

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-59})$$.
All geometric endomorphisms are defined over $\F_{3^{3}}$.

## 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 1.27.h $2$ 1.729.f