Properties

Label 2.169.abs_bfa
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:  $( 1 - 13 x )^{2}( 1 - 18 x + 169 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.256594102998$
Angle rank:  $1$ (numerical)
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21888 806529024 23292770811264 665416447679692800 19004925918315703563648 542800517580047118522525696 15502931912308550212098714123648 442779261620511949717910235394867200 12646218549000629892461926414396419609984 361188648080369222933585830417600853326178304

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 126 28238 4825710 815730526 137858217246 23298074272046 3937376159570574 665416605942610366 112455406918791204030 19004963774661792286478

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.aba $\times$ 1.169.as 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.ai_afa$2$(not in LMFDB)
2.169.i_afa$2$(not in LMFDB)
2.169.bs_bfa$2$(not in LMFDB)