Properties

Label 2.169.abt_bgi
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:  $1 - 45 x + 840 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.106377140183$, $\pm0.212100084678$
Angle rank:  $2$ (numerical)
Number field:  4.0.1357144.1
Galois group:  $D_{4}$
Jacobians:  42

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 42 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21752 805955104 23295483984512 665447215543159936 19005074541062408349752 542801012377710472811935744 15502933169872501333097205481592 442779264116447637015186472860480000 12646218552806027650997185100466681844352 361188648084800696640343234288592945466121504

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 125 28217 4826270 815768241 137859295325 23298095509742 3937376478961925 665416609693547233 112455406952630375150 19004963774894966827577

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.1357144.1.
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.bt_bgi$2$(not in LMFDB)