Properties

Label 2.7.af_q
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:  $1 - 5 x + 16 x^{2} - 35 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.169178782589$, $\pm0.473594973839$
Angle rank:  $2$ (numerical)
Number field:  4.0.57800.1
Galois group:  $D_{4}$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 26 2756 121784 5666336 284585886 13993468736 680297445854 33223427868800 1628029116513176 79793296617432996

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 57 354 2361 16933 118938 826059 5763153 40344078 282478897

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is 4.0.57800.1.
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.f_q$2$2.49.h_e