Properties

Label 2.199.aby_bnf
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:  $( 1 - 27 x + 199 x^{2} )( 1 - 23 x + 199 x^{2} )$
Frobenius angles:  $\pm0.0936959350875$, $\pm0.196619630811$
Angle rank:  $2$ (numerical)
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30621 1550065641 62088089585904 2459423877760491849 97393991078214868601901 3856888034022403438445097216 152736585098315399587521240270141 6048521417772032149087608069943787529 239527496521742950221530264785458129627504 9485528389643528757153063878549530712660313801

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39140 7878600 1568270884 312080606250 62103856973006 12358664467959750 2459374193055320644 489415464124909579800 97393677359686357343300

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The isogeny class factors as 1.199.abb $\times$ 1.199.ax 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.ae_aip$2$(not in LMFDB)
2.199.e_aip$2$(not in LMFDB)
2.199.by_bnf$2$(not in LMFDB)