# Properties

 Label 1.361.ai Base Field $\F_{19^{2}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19^{2}}$ Dimension: $1$ L-polynomial: $1 - 8 x + 361 x^{2}$ Frobenius angles: $\pm0.432482236283$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-345})$$ Galois group: $C_2$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 354 130980 47054034 16983390720 6131061936354 2213314946702820 799006687564020114 288441413571893422080 104127350297302429474914 37589973457539545413054500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 354 130980 47054034 16983390720 6131061936354 2213314946702820 799006687564020114 288441413571893422080 104127350297302429474914 37589973457539545413054500

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-345})$$.
All geometric endomorphisms are defined over $\F_{19^{2}}$.

## 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 1.361.i $2$ (not in LMFDB)