Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 2 x + 17 x^{2} )^{2}$ |
| $1 + 4 x + 38 x^{2} + 68 x^{3} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.577979130377$, $\pm0.577979130377$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $400$ | $102400$ | $23232400$ | $6922240000$ | $2022368410000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $350$ | $4726$ | $82878$ | $1424342$ | $24139550$ | $410258486$ | $6975884158$ | $118588986262$ | $2015989526750$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=9 x^6+14 x^4+14 x^2+9$
- $y^2=15 x^6+4 x^5+16 x^4+11 x^3+16 x^2+4 x+15$
- $y^2=13 x^6+11 x^4+11 x^2+13$
- $y^2=7 x^6+13 x^5+6 x^4+11 x^3+6 x^2+13 x+7$
- $y^2=x^6+7 x^5+7 x^4+4 x^3+11 x^2+10 x+13$
- $y^2=2 x^6+14 x^4+14 x^2+2$
- $y^2=2 x^5+9 x^4+11 x^3+9 x^2+2 x$
- $y^2=10 x^6+13 x^5+2 x^4+12 x^3+12 x^2+x+6$
- $y^2=13 x^6+16 x^5+3 x^4+6 x^3+3 x^2+16 x+13$
- $y^2=8 x^6+2 x^5+x^4+4 x^3+x^2+2 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.