Properties

Label 2.169.abx_bka
Base Field $\F_{13^{2}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 13 x )^{2}( 1 - 23 x + 169 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.154420958311$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21168 800743104 23274445630464 665386850963116800 19004938639139516899248 542800764409177865855287296 15502932786770818826863382108208 442779263493339365094039096398515200 12646218551154683096662435854401923574784 361188648078566017646134979180044812332915904

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 121 28033 4821910 815694241 137858309521 23298084866446 3937376381663209 665416608757128961 112455406937945936230 19004963774566911532753

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.aba $\times$ 1.169.ax 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_aka$2$(not in LMFDB)
2.169.d_aka$2$(not in LMFDB)
2.169.bx_bka$2$(not in LMFDB)
2.169.az_ma$3$(not in LMFDB)
2.169.ae_aja$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.ad_aka$2$(not in LMFDB)
2.169.d_aka$2$(not in LMFDB)
2.169.bx_bka$2$(not in LMFDB)
2.169.az_ma$3$(not in LMFDB)
2.169.ae_aja$3$(not in LMFDB)
2.169.abw_bja$6$(not in LMFDB)
2.169.abb_oa$6$(not in LMFDB)
2.169.e_aja$6$(not in LMFDB)
2.169.z_ma$6$(not in LMFDB)
2.169.bb_oa$6$(not in LMFDB)
2.169.bw_bja$6$(not in LMFDB)