Properties

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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon base change
2.19.l_ch$2$(not in LMFDB)