Properties

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

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 20 x + 188 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.110672799774$, $\pm0.337577521471$
Angle rank:  $2$ (numerical)
Number field:  4.0.5433600.1
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
$A(\F_{q^r})$ 1590 5708100 13881286710 33240526707600 79788566331849750 191579590025849204100 459986919375032320916310 1104428258654316787741593600 2651731107753872205214939352790 6366805820626796818077911187562500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2378 117990 5766118 282462150 13841168618 678223637310 33232948154878 1628413758738270 79792267046027498

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.5433600.1.
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_hg$2$(not in LMFDB)