Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 31 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.0673600794514$, $\pm0.932639920549$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{65})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $259$ | $67081$ | $24134656$ | $6912093321$ | $2015995373539$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $228$ | $4914$ | $82756$ | $1419858$ | $24131742$ | $410338674$ | $6975798148$ | $118587876498$ | $2015996846628$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=13 x^6+14 x^5+x^4+9 x^3+5 x^2+10 x+10$
- $y^2=16 x^6+2 x^5+7 x^4+14 x^3+8 x^2+4 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{2}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{65})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.abf 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-195}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.17.ad_u | $3$ | (not in LMFDB) |
| 2.17.d_u | $3$ | (not in LMFDB) |
| 2.17.a_bf | $4$ | (not in LMFDB) |