Properties

Label 2.169.abt_bgj
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:  $1 - 45 x + 841 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:  $\pm0.115648327740$, $\pm0.206920732394$
Angle rank:  $2$ (numerical)
Number field:  4.0.4901.1
Galois group:  $D_{4}$
Jacobians:  45

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 45 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21753 806013909 23296136610825 665451080819896869 19005090453458068800528 542801062270596059443963125 15502933290625536852775954242153 442779264323507656408335057141450309 12646218552930502563446620340855435746425 361188648084034268161062915899350798674038784

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 125 28219 4826405 815772979 137859410750 23298097651243 3937376509630325 665416610004720739 112455406953737257565 19004963774854639022254

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.4901.1.
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_bgj$2$(not in LMFDB)