Properties

Label 2.49.ax_iv
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:  $1 - 23 x + 229 x^{2} - 1127 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.142622734692$, $\pm0.234081986531$
Angle rank:  $2$ (numerical)
Number field:  4.0.81125.1
Galois group:  $D_{4}$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1481 5599661 13871225201 33266158087445 79806326823818496 191585122056539279501 459987182520552743279801 1104427667528199114470953445 2651730804376104684256767464081 6366805749648344848092123125846016

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 27 2331 117903 5770563 282525022 13841568291 678224025303 33232930367523 1628413572435627 79792266156487006

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.81125.1.
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.x_iv$2$(not in LMFDB)