Properties

Label 2.169.abt_bgh
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:  $1 - 45 x + 839 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.0974399185523$, $\pm0.216609910606$
Angle rank:  $2$ (numerical)
Number field:  4.0.7245189.1
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21751 805896301 23294831363239 665443347009001029 19005058566634093568176 542800961870456101271317621 15502933044905016862399804525951 442779263887007509261828388940321189 12646218552584336947644950532727540576759 361188648085211868841053499997619840786846976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 125 28215 4826135 815763499 137859179450 23298093341871 3937376447223155 665416609348740499 112455406950659009705 19004963774916601817550

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.7245189.1.
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_bgh$2$(not in LMFDB)