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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30100$ | $1544130000$ | $62057080783600$ | $2459308136954760000$ | $97393652036485928315500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $147$ | $38989$ | $7874664$ | $1568197081$ | $312079519857$ | $62103844215502$ | $12358664350762863$ | $2459374192189077841$ | $489415464115516684536$ | $97393677359420358532549$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=115x^6+80x^5+163x^4+184x^3+28x^2+10x+75$
- $y^2=166x^6+23x^5+106x^4+29x^3+15x^2+51x+155$
- $y^2=38x^6+120x^5+116x^4+174x^3+163x^2+61x+113$
- $y^2=134x^6+172x^5+106x^4+82x^3+99x^2+48x+67$
- $y^2=187x^6+107x^5+57x^4+150x^3+65x^2+67x+111$
- $y^2=13x^6+166x^5+48x^4+84x^3+81x^2+193x+108$
- $y^2=108x^6+69x^5+31x^4+93x^3+144x^2+167x+166$
- $y^2=135x^6+147x^5+153x^4+161x^3+140x^2+166x+117$
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.