Properties

Label 2.199.abx_bmb
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:  $1 - 49 x + 989 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0696172273604$, $\pm0.224914406157$
Angle rank:  $2$ (numerical)
Number field:  4.0.1912493.1
Galois group:  $D_{4}$
Jacobians:  $24$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $30791$ $1551589281$ $62091872695277$ $2459412645659510013$ $97393839515101513312016$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $151$ $39179$ $7879081$ $1568263723$ $312080120596$ $62103843747191$ $12358664218810207$ $2459374189577677699$ $489415464092084717173$ $97393677359613282430814$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 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 4.0.1912493.1.

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.bx_bmb$2$(not in LMFDB)