# Properties

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

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## Invariants

 Base field: $\F_{19^{2}}$ Dimension: $1$ L-polynomial: $1 - 37 x + 361 x^{2}$ Frobenius angles: $\pm0.0731863950403$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 325 129675 47035300 16983405075 6131064233125 2213314901179200 799006685851974925 288441413576634720675 104127350298219801213700 37589973457554121008916875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 325 129675 47035300 16983405075 6131064233125 2213314901179200 799006685851974925 288441413576634720675 104127350298219801213700 37589973457554121008916875

## 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 is a primitive isogeny class.

## Twists

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