Properties

Label 2.49.av_hu
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:  $( 1 - 13 x + 49 x^{2} )( 1 - 8 x + 49 x^{2} )$
Frobenius angles:  $\pm0.121037718324$, $\pm0.306389419005$
Angle rank:  $2$ (numerical)
Jacobians:  16

This isogeny class is not 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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1554 5678316 13885804296 33252581908224 79794396406737474 191580510300991209216 459986523882187819594434 1104427996894746505434184704 2651731067238129001644603234696 6366805840668972677984960081957676

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 29 2365 118028 5768209 282482789 13841235106 678223054181 33232940278369 1628413733857772 79792267297206925

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.an $\times$ 1.49.ai 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_ag$2$(not in LMFDB)
2.49.f_ag$2$(not in LMFDB)
2.49.v_hu$2$(not in LMFDB)
2.49.ag_de$3$(not in LMFDB)
2.49.d_k$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.af_ag$2$(not in LMFDB)
2.49.f_ag$2$(not in LMFDB)
2.49.v_hu$2$(not in LMFDB)
2.49.ag_de$3$(not in LMFDB)
2.49.d_k$3$(not in LMFDB)
2.49.at_he$6$(not in LMFDB)
2.49.ak_ek$6$(not in LMFDB)
2.49.ad_k$6$(not in LMFDB)
2.49.g_de$6$(not in LMFDB)
2.49.k_ek$6$(not in LMFDB)
2.49.t_he$6$(not in LMFDB)