Properties

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

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x )^{2}( 1 - 9 x + 49 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.277748883973$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1476 5573376 13830291216 33231254400000 79786022925180996 191576347731494240256 459984319459923999365316 1104426997383375729446400000 2651730714327564184694300053776 6366805750336303584917901627446016

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 27 2321 117558 5764513 282453147 13840934366 678219803883 33232910202433 1628413517137302 79792266165108881

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.ao $\times$ 1.49.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.af_abc$2$(not in LMFDB)
2.49.f_abc$2$(not in LMFDB)
2.49.x_iq$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.af_abc$2$(not in LMFDB)
2.49.f_abc$2$(not in LMFDB)
2.49.x_iq$2$(not in LMFDB)
2.49.aj_du$4$(not in LMFDB)
2.49.j_du$4$(not in LMFDB)