Properties

Label 2.49.aw_ii
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:  $1 - 22 x + 216 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.136516647719$, $\pm0.269709412854$
Angle rank:  $2$ (numerical)
Number field:  4.0.537408.1
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
$A(\F_{q^r})$ 1518 5643924 13885414686 33263775484368 79801769045878638 191582793961904169396 459986594183040608041182 1104427704360753091272864768 2651730915088328000444772803262 6366805805600614975128592004413044

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2350 118024 5770150 282508888 13841400094 678223157836 33232931475838 1628413640423404 79792266857711230

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.537408.1.
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.w_ii$2$(not in LMFDB)