Invariants
| Base field: | $\F_{163}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 25 x + 163 x^{2} )( 1 - 23 x + 163 x^{2} )$ |
| $1 - 48 x + 901 x^{2} - 7824 x^{3} + 26569 x^{4}$ | |
| Frobenius angles: | $\pm0.0652307277549$, $\pm0.143017980409$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19599$ | $692687457$ | $18736672536144$ | $498294237640546809$ | $13239644246156138621199$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $116$ | $26068$ | $4326428$ | $705887428$ | $115063688996$ | $18755374494598$ | $3057125336771804$ | $498311415684461956$ | $81224760549897535844$ | $13239635967174192988468$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=32x^6+16x^5+68x^4+36x^3+162x^2+22x+124$
- $y^2=35x^6+88x^5+59x^4+94x^3+48x^2+39x+41$
- $y^2=137x^6+154x^5+4x^4+161x^3+40x^2+78x+80$
- $y^2=72x^6+125x^5+47x^4+88x^3+26x^2+142x+89$
- $y^2=96x^6+33x^5+25x^4+97x^3+151x^2+150x+161$
- $y^2=48x^6+26x^5+35x^4+86x^3+131x^2+22x+123$
- $y^2=6x^6+159x^5+58x^4+8x^3+58x^2+159x+6$
- $y^2=159x^6+113x^5+66x^4+75x^3+52x^2+161x+120$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163}$.
Endomorphism algebra over $\F_{163}$| The isogeny class factors as 1.163.az $\times$ 1.163.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.