Properties

Label 2.199.aby_bnc
Base field $\F_{199}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 50 x + 1016 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0639639324574$, $\pm0.208870266773$
Angle rank:  $2$ (numerical)
Number field:  4.0.90944.1
Galois group:  $D_{4}$
Jacobians:  $36$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $30618$ $1549821924$ $62084538832050$ $2459395976883917904$ $97393835662033816874178$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $150$ $39134$ $7878150$ $1568253094$ $312080108250$ $62103846002798$ $12358664269243050$ $2459374190063741374$ $489415464088360906950$ $97393677359360074887854$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{199}$.

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.90944.1.

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_bnc$2$(not in LMFDB)