# Properties

 Label 1.331.abg 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: $1 - 32 x + 331 x^{2}$ Frobenius angles: $\pm0.157917618260$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 10

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 10 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 300 109200 36263700 12003700800 3973198957500 1315127884870800 435307307458577700 144086718372002515200 47692703775861198420300 15786284949772700784330000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 300 109200 36263700 12003700800 3973198957500 1315127884870800 435307307458577700 144086718372002515200 47692703775861198420300 15786284949772700784330000

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{331}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.331.bg $2$ (not in LMFDB) 1.331.b $3$ (not in LMFDB) 1.331.bf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.331.bg $2$ (not in LMFDB) 1.331.b $3$ (not in LMFDB) 1.331.bf $3$ (not in LMFDB) 1.331.abf $6$ (not in LMFDB) 1.331.ab $6$ (not in LMFDB)