Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 47 x^{2} + 136 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.588687536289$, $\pm0.744645797044$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $481$ | $92833$ | $23136100$ | $7002671689$ | $2017140378721$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $320$ | $4706$ | $83844$ | $1420666$ | $24135590$ | $410332858$ | $6975694084$ | $118588692482$ | $2015991713600$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+16 x^5+7 x^4+12 x^3+16 x^2+x+9$
- $y^2=x^6+x^5+10 x^4+7 x^3+9 x+4$
- $y^2=16 x^6+4 x^5+6 x^4+2 x^3+13 x^2+14 x+8$
- $y^2=13 x^6+5 x^5+6 x^4+2 x^3+5 x^2+2$
- $y^2=2 x^6+10 x^5+7 x^4+16 x^3+8 x^2+14 x+9$
- $y^2=8 x^6+11 x^5+7 x^4+11 x^3+9 x^2+12 x+2$
- $y^2=16 x^6+16 x^5+8 x^4+8 x^3+10 x^2+10 x+1$
- $y^2=4 x^6+16 x^5+7 x^4+13 x^2+16 x+4$
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 \(\Q(\zeta_{12})\). |
| The base change of $A$ to $\F_{17^{3}}$ is 1.4913.aea 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.