Properties

Label 2.199.abz_bog
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:  $( 1 - 27 x + 199 x^{2} )( 1 - 24 x + 199 x^{2} )$
Frobenius angles:  $\pm0.0936959350875$, $\pm0.176204172288$
Angle rank:  $2$ (numerical)
Jacobians:  16

This isogeny class is not 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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30448 1548219904 62079739927744 2459401102663464960 97393967494055688776848 3856888170158997044063297536 152736586098917989578622601182288 6048521421685204791143189561577431040 239527496532198154018446454847219651164864 9485528389659396058259373625856413406510831104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 149 39093 7877540 1568256361 312080530679 62103859165086 12358664548923401 2459374194646446001 489415464146272214540 97393677359849276554653

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The isogeny class factors as 1.199.abb $\times$ 1.199.ay and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ad_ajq$2$(not in LMFDB)
2.199.d_ajq$2$(not in LMFDB)
2.199.bz_bog$2$(not in LMFDB)