Properties

Label 2.49.aw_ij
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 22 x + 217 x^{2} - 1078 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.152971575916$, $\pm0.259933654116$
Angle rank:  $2$ (numerical)
Number field:  4.0.5696.1
Galois group:  $D_{4}$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 1519 5649161 13893241852 33269948476121 79804876484494399 191583774042743125904 459986683798301467340239 1104427595277195005741719529 2651730831957367561904046208252 6366805770547514995544341769190201

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2352 118090 5771220 282519888 13841470902 678223289968 33232928193444 1628413589373130 79792266418406752

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.5696.1.
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_ij$2$(not in LMFDB)