Properties

Label 2.19.al_cg
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:  $1 - 11 x + 58 x^{2} - 209 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0194296767276$, $\pm0.415073966325$
Angle rank:  $2$ (numerical)
Number field:  4.0.8405.1
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
$A(\F_{q^r})$ 200 128000 46738400 16847360000 6116072751000 2212693071872000 799009152438764600 288439869271132160000 104126790334174227960800 37589921557333848835200000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 357 6816 129273 2470039 47032662 893874501 16983472113 322685962464 6131057792677

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
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_cg$2$(not in LMFDB)