Properties

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

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 13 x )^{2}( 1 - 22 x + 169 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.178912375022$
Angle rank:  $1$ (numerical)
Jacobians:  20

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 20 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21312 802013184 23279325915456 665399220653260800 19004958209563460165952 542800764409177865855287296 15502932655305798311356598344512 442779262923730269141293513519923200 12646218549535878102276395869049620789056 361188648075159905971438065308456186141589504

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 122 28078 4822922 815709406 137858451482 23298084866446 3937376348274218 665416607901110206 112455406923550852538 19004963774387689319278

Decomposition and endomorphism algebra

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