Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 17 x^{2} )^{2}$ |
| $1 + 34 x^{2} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $14$ |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $324$ | $104976$ | $24147396$ | $6879707136$ | $2015996740164$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $358$ | $4914$ | $82366$ | $1419858$ | $24157222$ | $410338674$ | $6975423358$ | $118587876498$ | $2015999579878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=5 x^6+9 x^5+14 x^4+6 x^3+12 x^2+8 x+3$
- $y^2=15 x^6+10 x^5+8 x^4+x^3+2 x^2+7 x+9$
- $y^2=6 x^6+6 x^5+13 x^4+3 x^3+13 x^2+6 x+6$
- $y^2=x^6+x^5+5 x^4+9 x^3+5 x^2+x+1$
- $y^2=x^6+x^3+12$
- $y^2=x^6+16$
- $y^2=3 x^6+14$
- $y^2=x^6+7$
- $y^2=x^6+11 x^5+3 x^4+7 x^3+14 x^2+11 x+16$
- $y^2=x^6+x^3+9$
- $y^2=3 x^6+3 x^3+10$
- $y^2=13 x^6+6 x^5+5 x^4+10 x^3+6 x^2+10 x+9$
- $y^2=5 x^6+x^5+15 x^4+13 x^3+x^2+13 x+10$
- $y^2=16 x^6+9 x^5+5 x^4+15 x^2+4 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{2}}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$ |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.bi 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $17$ and $\infty$. |
Base change
This is a primitive isogeny class.