Properties

Label 2.49.aw_if
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:  $1 - 22 x + 213 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0895606637112$, $\pm0.290867746445$
Angle rank:  $2$ (numerical)
Number field:  4.0.1069632.1
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1515 5628225 13861940940 33245120238825 79792074188073075 191579344836288051600 459985860344438202800595 1104427717001232941391796425 2651731001171207920385266455180 6366805846821203728669412351030625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2344 117826 5766916 282474568 13841150902 678222075832 33232931856196 1628413693286434 79792267374310024

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.1069632.1.
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.w_if$2$(not in LMFDB)