Properties

Label 2.169.abw_bjd
Base Field $\F_{13^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x + 169 x^{2} )( 1 - 23 x + 169 x^{2} )$
Frobenius angles:  $\pm0.0885687144757$, $\pm0.154420958311$
Angle rank:  $2$ (numerical)
Jacobians:  14

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 14 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21315 802190025 23281414813440 665412853037812425 19005022727784799137075 542801011479319554180710400 15502933462773514450484042880195 442779265240398577491531563986050825 12646218555456539121727675671671789633280 361188648088711714489064028737538772038875625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 122 28084 4823354 815726116 137858919482 23298095471182 3937376553351818 665416611382640836 112455406976199819626 19004963775100756109524

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.az $\times$ 1.169.ax 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 $\F_{13^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.ac_ajd$2$(not in LMFDB)
2.169.c_ajd$2$(not in LMFDB)
2.169.bw_bjd$2$(not in LMFDB)
2.169.ay_mb$3$(not in LMFDB)
2.169.ad_aie$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.ac_ajd$2$(not in LMFDB)
2.169.c_ajd$2$(not in LMFDB)
2.169.bw_bjd$2$(not in LMFDB)
2.169.ay_mb$3$(not in LMFDB)
2.169.ad_aie$3$(not in LMFDB)
2.169.abv_bie$6$(not in LMFDB)
2.169.aba_nz$6$(not in LMFDB)
2.169.d_aie$6$(not in LMFDB)
2.169.y_mb$6$(not in LMFDB)
2.169.ba_nz$6$(not in LMFDB)
2.169.bv_bie$6$(not in LMFDB)