# Properties

 Label 1.361.aba 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: $1 - 26 x + 361 x^{2}$ Frobenius angles: $\pm0.260146938293$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 336 130368 47056464 16983821568 6131069159376 2213314901179200 799006684270354896 288441413534752601088 104127350297602681851984 37589973457549801194193728

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 336 130368 47056464 16983821568 6131069159376 2213314901179200 799006684270354896 288441413534752601088 104127350297602681851984 37589973457549801194193728

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
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.ai $\F_{19}$ 1.19.i

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.361.ba $2$ (not in LMFDB) 1.361.al $3$ (not in LMFDB) 1.361.bl $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.361.ba $2$ (not in LMFDB) 1.361.al $3$ (not in LMFDB) 1.361.bl $3$ (not in LMFDB) 1.361.abl $6$ (not in LMFDB) 1.361.l $6$ (not in LMFDB)