Properties

Label 2.169.abt_bge
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:  $1 - 45 x + 836 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0700871878856$, $\pm0.227679424698$
Angle rank:  $2$ (numerical)
Number field:  4.0.2552616.2
Galois group:  $D_{4}$
Jacobians:  18

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 18 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21748 805719904 23292873529600 665431721861791104 19005010271152912121428 542800806659684204600320000 15502932644641453119946286344468 442779263063380882416378908470040064 12646218551326233601779281646726511878400 361188648084242447368810060036397133964569184

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 125 28209 4825730 815749249 137858829125 23298086679918 3937376345565725 665416608110979649 112455406939471434290 19004963774865592960929

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.2552616.2.
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_bge$2$(not in LMFDB)