Properties

Label 2.7.ae_k
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:  $1 - 4 x + 10 x^{2} - 28 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.134159304295$, $\pm0.550039816438$
Angle rank:  $2$ (numerical)
Number field:  4.0.2048.2
Galois group:  $C_4$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 28 2576 108892 5605376 287844508 13955163152 678356898268 33251359490048 1629327895324444 79796787595788816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 54 316 2334 17124 118614 823708 5767998 40376260 282491254

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is 4.0.2048.2.
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.e_k$2$2.49.e_aba