Properties

Label 2.19.aj_br
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:  $1 - 9 x + 43 x^{2} - 171 x^{3} + 361 x^{4}$
Frobenius angles:  $\pm0.0855093505514$, $\pm0.478262641089$
Angle rank:  $2$ (numerical)
Number field:  4.0.18605.1
Galois group:  $D_{4}$
Jacobians:  15

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 15 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 225 131625 46500075 16848658125 6126704118000 2213933438354625 799047563932652175 288440050825745983125 104127413577379176804525 37590024281049051458400000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 11 367 6779 129283 2474336 47059027 893917469 16983482803 322687893881 6131074547302

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 4.0.18605.1.
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.j_br$2$(not in LMFDB)