Properties

Label 2.169.abu_bhd
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:  $1 - 46 x + 861 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0656191264145$, $\pm0.209870530721$
Angle rank:  $2$ (numerical)
Number field:  4.0.4138560.2
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21603 804560529 23289207215724 665425772794246665 19005012280938171649323 542800851025004646111263376 15502932787607947656559369989723 442779263283147485263646341235045385 12646218551158784464590707822505288940524 361188648082010635211423581489244550958282929

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 124 28168 4824970 815741956 137858843704 23298088584166 3937376381875816 665416608441248836 112455406937982407290 19004963774748159842728

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.4138560.2.
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.bu_bhd$2$(not in LMFDB)