Properties

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

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $( 1 - 28 x + 199 x^{2} )( 1 - 25 x + 199 x^{2} )$
  $1 - 53 x + 1098 x^{2} - 10547 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0391815390403$, $\pm0.153403448314$
Angle rank:  $2$ (numerical)
Jacobians:  $8$

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $30100$ $1544130000$ $62057080783600$ $2459308136954760000$ $97393652036485928315500$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $147$ $38989$ $7874664$ $1568197081$ $312079519857$ $62103844215502$ $12358664350762863$ $2459374192189077841$ $489415464115516684536$ $97393677359420358532549$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 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 isogeny class factors as 1.199.abc $\times$ 1.199.az and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.199.ad_alq$2$(not in LMFDB)
2.199.d_alq$2$(not in LMFDB)
2.199.cb_bqg$2$(not in LMFDB)
2.199.ao_et$3$(not in LMFDB)
2.199.ai_abb$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.199.ad_alq$2$(not in LMFDB)
2.199.d_alq$2$(not in LMFDB)
2.199.cb_bqg$2$(not in LMFDB)
2.199.ao_et$3$(not in LMFDB)
2.199.ai_abb$3$(not in LMFDB)
2.199.abq_bfr$6$(not in LMFDB)
2.199.abk_zx$6$(not in LMFDB)
2.199.i_abb$6$(not in LMFDB)
2.199.o_et$6$(not in LMFDB)
2.199.bk_zx$6$(not in LMFDB)
2.199.bq_bfr$6$(not in LMFDB)