Properties

Label 2.49.au_hf
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:  $1 - 20 x + 187 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0998650046166$, $\pm0.341585099009$
Angle rank:  $2$ (numerical)
Number field:  4.0.5469200.1
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
$A(\F_{q^r})$ 1589 5702921 13874188244 33235625576825 79786476121393629 191579012411680107536 459986815688615935867469 1104428239264561746233558825 2651731098559946289403790934164 6366805817766065145782928634039641

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2376 117930 5765268 282454750 13841126886 678223484430 33232947571428 1628413753092330 79792267010175256

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.5469200.1.
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_hf$2$(not in LMFDB)