Properties

Label 2.169.abu_bhe
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:  $1 - 46 x + 862 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0773526067278$, $\pm0.205567582789$
Angle rank:  $2$ (numerical)
Number field:  4.0.239600.1
Galois group:  $D_{4}$
Jacobians:  36

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 36 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21604 804619376 23289874383556 665429866510601984 19005030100615640363524 542800912180161456754574576 15502932961009768024562587723684 442779263696544139697640219858366464 12646218551983479367325434378846818366436 361188648083340125983607193574447350875195376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 124 28170 4825108 815746974 137858972964 23298091209066 3937376425915756 665416609062508734 112455406945315935532 19004963774818114765450

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.239600.1.
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.bu_bhe$2$(not in LMFDB)