Properties

Label 2.49.au_hc
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:  $1 - 20 x + 184 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0612448027520$, $\pm0.352483684171$
Angle rank:  $2$ (numerical)
Number field:  4.0.3524864.2
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1586 5687396 13852899554 33220785273104 79779866673449826 191576867326707778916 459986174198699130180146 1104427985589086385272274944 2651730979534664302874014749554 6366805772990825843560346750130276

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2370 117750 5762694 282431350 13840971906 678222538590 33232939938174 1628413679999550 79792266449027650

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.3524864.2.
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_hc$2$(not in LMFDB)