Properties

Label 2.169.abv_bib
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:  $1 - 47 x + 885 x^{2} - 7943 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0403592669200$, $\pm0.196341187291$
Angle rank:  $2$ (numerical)
Number field:  4.0.1078245.4
Galois group:  $D_{4}$
Jacobians:  8

This isogeny class is simple and geometrically 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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21457 803285709 23284251363121 665412276240794805 19004983497851678762752 542800798102086644107233981 15502932680682468026291308989577 442779262964202189118624972009596645 12646218549962334907166403361053079632121 361188648077763136031833015059895168261730304

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 123 28123 4823943 815725411 137858634918 23298086312611 3937376354719287 665416607961932131 112455406927343082867 19004963774524665642478

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.1078245.4.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.bv_bib$2$(not in LMFDB)