# Properties

 Label 2.19.al_ch Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $1 - 11 x + 59 x^{2} - 209 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0641455582635$, $\pm0.408994580500$ Angle rank: $2$ (numerical) Number field: 4.0.291597.2 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

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

• $y^2=2x^6+5x^5+5x^4+10x^3+10x^2+5x+15$
• $y^2=3x^6+4x^5+5x^4+4x^3+18x^2+15x+12$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 201 128841 46967067 16879845933 6119201069616 2212993652334129 799046735188015263 288444503541941961237 104127241103131388139333 37589957352536821973926656

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 359 6849 129523 2471304 47039051 893916543 16983744979 322687359387 6131063631014

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.291597.2.
All geometric endomorphisms are defined over $\F_{19}$.

## 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 2.19.l_ch $2$ (not in LMFDB)