Properties

Label 2.49.au_hk
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:  $1 - 20 x + 192 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.151227437545$, $\pm0.318680514787$
Angle rank:  $2$ (numerical)
Number field:  4.0.3283200.1
Galois group:  $D_{4}$
Jacobians:  16

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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1594 5728836 13909691914 33259903165200 79796362518183754 191581217848125747396 459986798595930939118714 1104428036413689150823219200 2651731021732203448287443867194 6366805796707310078942043343032516

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2386 118230 5769478 282489750 13841286226 678223459230 33232941467518 1628413705912830 79792266746255506

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.3283200.1.
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.u_hk$2$(not in LMFDB)