Properties

Label 2.199.aby_bmz
Base Field $\F_{199}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 50 x + 1013 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0191784432260$, $\pm0.218244196354$
Angle rank:  $2$ (numerical)
Number field:  4.0.1473600.1
Galois group:  $D_{4}$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30615 1549578225 62080988133060 2459368019623545225 97393678841553823350375 3856886653100608621553763600 152736580019093463529344791624535 6048521401857953932435504380335721225 239527496478819655379817919394917857743940 9485528389543214778777981047643795186450140625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39128 7877700 1568235268 312079605750 62103834737318 12358664056975050 2459374186584537028 489415464037206395100 97393677358656372840248

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.1473600.1.
All geometric endomorphisms are defined over $\F_{199}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.199.by_bmz$2$(not in LMFDB)