Properties

Label 2.49.au_hd
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:  $1 - 20 x + 185 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0757393158182$, $\pm0.349014705192$
Angle rank:  $2$ (numerical)
Number field:  4.0.285441.2
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
$A(\F_{q^r})$ 1587 5692569 13859994672 33225754864041 79782126293609427 191577651228456931584 459986443670183154118707 1104428103711388769740775625 2651731035572970617762997135792 6366805794911293582364021872918329

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2372 117810 5763556 282439350 13841028542 678222935910 33232943492548 1628413714412370 79792266723746852

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.285441.2.
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.u_hd$2$(not in LMFDB)