Properties

Label 2.169.aby_blb
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:  $( 1 - 25 x + 169 x^{2} )^{2}$
Frobenius angles:  $\pm0.0885687144757$, $\pm0.0885687144757$
Angle rank:  $1$ (numerical)
Jacobians:  6

This isogeny class is not 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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21025 799475625 23269625299600 665375421944975625 19004927208209142600625 542800814692874264616960000 15502933166409639470199270448225 442779265099756037591146070376575625 12646218556563926064314771702828281728400 361188648094342857041324591376435597266015625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 120 27988 4820910 815680228 137858226600 23298087024718 3937376478082440 665416611171280708 112455406986047162430 19004963775397054624948

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

SubfieldPrimitive Model
$\F_{13}$2.13.a_az

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_alb$2$(not in LMFDB)
2.169.by_blb$2$(not in LMFDB)
2.169.z_ro$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_alb$2$(not in LMFDB)
2.169.by_blb$2$(not in LMFDB)
2.169.z_ro$3$(not in LMFDB)
2.169.a_lb$4$(not in LMFDB)
2.169.az_ro$6$(not in LMFDB)