Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 32 x^{2} + 136 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.490632536990$, $\pm0.990632536990$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{8})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 3 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $466$ | $82948$ | $24896050$ | $6880370704$ | $2018846991346$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $290$ | $5066$ | $82374$ | $1421866$ | $24137570$ | $410386618$ | $6975432574$ | $118588360922$ | $2015993900450$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=x^5+14 x$
- $y^2=x^6+3 x^5+2 x^4+2 x^2+14 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{4}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |
| The base change of $A$ to $\F_{17^{4}}$ is 1.83521.awc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{17^{2}}$
The base change of $A$ to $\F_{17^{2}}$ is the simple isogeny class 2.289.a_awc and its endomorphism algebra is \(\Q(\zeta_{8})\).
Base change
This is a primitive isogeny class.