# Properties

 Label 1.331.abk Base Field $\F_{331}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{331}$ Dimension: $1$ L-polynomial: $1 - 36 x + 331 x^{2}$ Frobenius angles: $\pm0.0464545124559$ Angle rank: $1$ (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 ordinary.

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

## 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 $A(\F_{q^r})$ 296 108928 36253784 12003429888 3973192839176 1315127766870400 435307305521909816 144086718346331977728 47692703775643094106344 15786284949773773767046528

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 296 108928 36253784 12003429888 3973192839176 1315127766870400 435307305521909816 144086718346331977728 47692703775643094106344 15786284949773773767046528

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{331}$.

## 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 1.331.bk $2$ (not in LMFDB)