Properties

Label 2.169.abs_bfd
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:  $1 - 44 x + 809 x^{2} - 7436 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0555167869616$, $\pm0.249832489919$
Angle rank:  $2$ (numerical)
Number field:  4.0.1161537.2
Galois group:  $D_{4}$
Jacobians:  32

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 32 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21891 806705241 23294684894832 665427495056297193 19004970501848006305491 542800658486854238356963584 15502932285151972432420160100003 442779262501694442387055234308047817 12646218551033267322655977577992660308208 361188648085407499853094412496076478447777881

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 126 28244 4826106 815744068 137858540646 23298080320046 3937376254263942 665416607266867204 112455406936866256938 19004963774926895497364

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.1161537.2.
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.bs_bfd$2$(not in LMFDB)