Properties

Label 2.169.abw_bjd
Base field $\F_{13^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x + 169 x^{2} )( 1 - 23 x + 169 x^{2} )$
  $1 - 48 x + 913 x^{2} - 8112 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0885687144757$, $\pm0.154420958311$
Angle rank:  $2$ (numerical)
Jacobians:  $14$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $21315$ $802190025$ $23281414813440$ $665412853037812425$ $19005022727784799137075$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

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:

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)