Properties

Label 2.169.abt_bgf
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:  $1 - 45 x + 837 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0795349314068$, $\pm0.224304217515$
Angle rank:  $2$ (numerical)
Number field:  4.0.9860725.1
Galois group:  $D_{4}$
Jacobians:  40

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 40 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21749 805778701 23293526135789 665435600168336725 19005026431679376465104 542800859011722574225492621 15502932782296934417995655463269 442779263360576723244600296641642725 12646218551845414665141604606144941337709 361188648084939903511598165800956803748130816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 125 28211 4825865 815754003 137858946350 23298088926971 3937376380526945 665416608557610883 112455406944088207085 19004963774902291591406

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.9860725.1.
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_bgf$2$(not in LMFDB)