Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 16 x^{2} + 17 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.205362651562$, $\pm0.872029318229$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-67})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $292$ | $74752$ | $24641296$ | $7018614784$ | $2017926759652$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $19$ | $257$ | $5014$ | $84033$ | $1421219$ | $24152222$ | $410308211$ | $6975851521$ | $118586652598$ | $2015992913057$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=2 x^6+13 x^5+16 x^4+16 x^3+6 x^2+11 x+2$
- $y^2=11 x^6+14 x^5+11 x^4+2 x^3+16 x^2+5 x+16$
- $y^2=9 x^5+13 x^4+10 x^3+x^2+3 x+11$
- $y^2=2 x^6+13 x^5+7 x^4+5 x^3+10 x^2+14$
- $y^2=15 x^5+2 x^4+8 x^3+3 x^2+x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{3}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-67})\). |
| The base change of $A$ to $\F_{17^{3}}$ is 1.4913.by 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$ |
Base change
This is a primitive isogeny class.