Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 3 x + 17 x^{2}$ |
| Frobenius angles: | $\pm0.618522015261$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-59}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21$ | $315$ | $4788$ | $83475$ | $1422141$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $315$ | $4788$ | $83475$ | $1422141$ | $24131520$ | $410318013$ | $6975922275$ | $118587733236$ | $2015991528075$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 0 are hyperelliptic):
- $y^2=x^3+15 x+15$
- $y^2=x^3+5 x+5$
- $y^2=x^3+10 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-59}) \). |
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 |
|---|---|---|
| 1.17.ad | $2$ | 1.289.z |