Properties

Label 2.49.ax_ir
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 23 x + 225 x^{2} - 1127 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0550320885809$, $\pm0.271501929914$
Angle rank:  $2$ (numerical)
Number field:  4.0.284445.2
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1477 5578629 13838475301 33238280483205 79790213515005952 191578291100868522501 459985079447809912612357 1104427274187618598029610245 2651730820716027483375496370341 6366805796193014813228463108194304

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 27 2323 117627 5765731 282467982 13841074771 678220924443 33232918531651 1628413582469883 79792266739810078

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.284445.2.
All geometric endomorphisms are defined over $\F_{7^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.x_ir$2$(not in LMFDB)