# Properties

 Label 1.361.a Base Field $\F_{19^{2}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19^{2}}$ Dimension: $1$ L-polynomial: $1 + 361 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 362 131044 47045882 16983302400 6131066257802 2213315013157924 799006685782884122 288441413533654041600 104127350297911241532842 37589973457558220325871204

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 362 131044 47045882 16983302400 6131066257802 2213315013157924 799006685782884122 288441413533654041600 104127350297911241532842 37589973457558220325871204

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{19^{2}}$
 The base change of $A$ to $\F_{19^{4}}$ is the simple isogeny class 1.130321.bbu and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $19$ and $\infty$.
All geometric endomorphisms are defined over $\F_{19^{4}}$.

## 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.abm $4$ (not in LMFDB) 1.361.bm $4$ (not in LMFDB)