Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.131754293880$, $\pm0.868245706120$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-11}, \sqrt{57})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $267$ | $71289$ | $24145344$ | $6984112041$ | $2015995440507$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $244$ | $4914$ | $83620$ | $1419858$ | $24153118$ | $410338674$ | $6976086724$ | $118587876498$ | $2015996980564$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=7 x^6+7 x^5+16 x^4+13 x^3+3 x^2+7 x+11$
- $y^2=4 x^6+4 x^5+14 x^4+5 x^3+9 x^2+4 x+16$
- $y^2=14 x^6+4 x^5+4 x^4+x^3+8 x^2+6 x+5$
- $y^2=11 x^6+14 x^5+9 x^4+8 x^3+8 x^2+9 x+11$
- $y^2=11 x^6+3 x^5+15 x^4+7 x^2+13 x+1$
- $y^2=16 x^6+9 x^5+11 x^4+4 x^2+5 x+3$
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 \(\Q(\sqrt{-11}, \sqrt{57})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-627}) \)$)$ |
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.a_x | $4$ | (not in LMFDB) |