# Properties

 Label 1.361.c 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: $1 + 2 x + 361 x^{2}$ Frobenius angles: $\pm0.516760896166$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-10})$$ Galois group: $C_2$ Jacobians: 30

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 30 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 364 131040 47043724 16983308160 6131067546604 2213315008500960 799006685138757004 288441413536623459840 104127350298205766838124 37589973457556559315276000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 364 131040 47043724 16983308160 6131067546604 2213315008500960 799006685138757004 288441413536623459840 104127350298205766838124 37589973457556559315276000

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-10})$$.
All geometric endomorphisms are defined over $\F_{19^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{19^{2}}$.

 Subfield Primitive Model $\F_{19}$ 1.19.ag $\F_{19}$ 1.19.g

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.361.ac $2$ (not in LMFDB)