# Properties

 Label 1.19.ab Base Field $\F_{19}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $1$ L-polynomial: $1 - x + 19 x^{2}$ Frobenius angles: $\pm0.463406802480$ 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 $A(\F_{q^r})$ 19 399 6916 129675 2474389 47056464 893914831 16983405075 322686721084 6131068282479

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 19 399 6916 129675 2474389 47056464 893914831 16983405075 322686721084 6131068282479

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{19}$.

## 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 1.19.b $2$ 1.361.bl 1.19.ah $3$ (not in LMFDB) 1.19.i $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.19.b $2$ 1.361.bl 1.19.ah $3$ (not in LMFDB) 1.19.i $3$ (not in LMFDB) 1.19.ai $6$ (not in LMFDB) 1.19.h $6$ (not in LMFDB)