Properties

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

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$
Frobenius angles:  $\pm0.0885687144757$, $\pm0.178912375022$
Angle rank:  $2$ (numerical)
Jacobians:  22

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 22 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21460 803462400 23286296559760 665425223211340800 19005042298295333077300 542801011479319554180710400 15502933331308488202467838412980 442779264670789479291298574632243200 12646218553837734126790967833290057669520 361188648085305602814271438059249622659360000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 123 28129 4824366 815741281 137859061443 23298095471182 3937376519962827 665416610526622081 112455406961804735934 19004963774921533896049

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.az $\times$ 1.169.aw 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.ad_aie$2$(not in LMFDB)
2.169.d_aie$2$(not in LMFDB)
2.169.bv_bie$2$(not in LMFDB)
2.169.aba_nz$3$(not in LMFDB)
2.169.ac_ajd$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.ad_aie$2$(not in LMFDB)
2.169.d_aie$2$(not in LMFDB)
2.169.bv_bie$2$(not in LMFDB)
2.169.aba_nz$3$(not in LMFDB)
2.169.ac_ajd$3$(not in LMFDB)
2.169.abw_bjd$6$(not in LMFDB)
2.169.ay_mb$6$(not in LMFDB)
2.169.c_ajd$6$(not in LMFDB)
2.169.y_mb$6$(not in LMFDB)
2.169.ba_nz$6$(not in LMFDB)
2.169.bw_bjd$6$(not in LMFDB)