Properties

Label 2.7.ah_z
Base Field $\F_{7}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7}$
Dimension:  $2$
L-polynomial:  $1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.162349854003$, $\pm0.351370772325$
Angle rank:  $2$ (numerical)
Number field:  4.0.3725.1
Galois group:  $D_{4}$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 19 2489 130321 5946221 282939184 13845433361 679394596669 33272252045909 1628923563606439 79788265901524224

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 51 379 2475 16836 117687 824965 5771619 40366243 282461086

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is 4.0.3725.1.
All geometric endomorphisms are defined over $\F_{7}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.7.h_z$2$2.49.b_bl