Invariants
| Base field: | $\F_{199}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 28 x + 199 x^{2} )( 1 - 23 x + 199 x^{2} )$ |
| $1 - 51 x + 1042 x^{2} - 10149 x^{3} + 39601 x^{4}$ | |
| Frobenius angles: | $\pm0.0391815390403$, $\pm0.196619630811$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30444$ | $1547894736$ | $62074910607696$ | $2459362031800974144$ | $97393740848463325045524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $149$ | $39085$ | $7876928$ | $1568231449$ | $312079804439$ | $62103842259406$ | $12358664218598441$ | $2459374189105787089$ | $489415464066160350272$ | $97393677358866325188325$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=195x^6+72x^5+91x^4+173x^3+x^2+143x+72$
- $y^2=59x^5+104x^4+94x^3+142x^2+191x+28$
- $y^2=96x^6+136x^5+31x^4+98x^3+54x^2+124x+176$
- $y^2=145x^6+120x^5+165x^3+157x^2+178x+34$
- $y^2=47x^6+26x^5+134x^4+150x^3+27x+3$
- $y^2=108x^6+192x^5+189x^4+138x^3+92x^2+36x+76$
- $y^2=121x^6+175x^5+106x^4+52x^3+12x^2+105x+85$
- $y^2=191x^6+66x^5+166x^4+27x^3+23x^2+129x+147$
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.ax 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.