Properties

Label 2.49.au_hj
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:  $1 - 20 x + 191 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.141161076953$, $\pm0.323951802917$
Angle rank:  $2$ (numerical)
Number field:  4.0.446096.4
Galois group:  $D_{4}$
Jacobians:  18

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 18 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1593 5723649 13902588900 33255093251529 79794498166934553 191580912954954810000 459986907979936779958233 1104428134996072196481567369 2651731059461212841968660040100 6366805806173992134648374435959809

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2384 118170 5768644 282483150 13841264198 678223620510 33232944433924 1628413729082010 79792266864897104

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.446096.4.
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_hj$2$(not in LMFDB)