Properties

Label 2.199.abx_bmj
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:  $1 - 49 x + 997 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.137603390886$, $\pm0.189051491022$
Angle rank:  $2$ (numerical)
Number field:  4.0.870725.2
Galois group:  $D_{4}$
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30799 1552238801 62101151528581 2459483308781640029 97394215085391547199824 3856888745689407261763893161 152736586942292009039093086538329 6048521421106862333214830393938462709 239527496521934978650202205080459408333611 9485528389610226443175235703187279137785502976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 151 39195 7880257 1568308779 312081324036 62103868432311 12358664617164919 2459374194411287619 489415464125301942613 97393677359344422272350

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.870725.2.
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.bx_bmj$2$(not in LMFDB)