Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 29 x^{2} - 102 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.195333930750$, $\pm0.528667264083$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $211$ | $90097$ | $24145996$ | $6967831689$ | $2020846600651$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $312$ | $4914$ | $83428$ | $1423272$ | $24154422$ | $410331192$ | $6975599236$ | $118587876498$ | $2015992855032$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=10 x^6+10 x^5+x^4+5 x^3+8 x^2+15 x+6$
- $y^2=12 x^6+2 x^5+9 x^4+2 x^3+16 x^2+8 x+10$
- $y^2=6 x^6+15 x^5+11 x^4+13 x^3+6 x^2+13 x+15$
- $y^2=12 x^6+11 x^5+x^4+12 x^3+6 x^2+4 x+3$
- $y^2=3 x^6+12 x^5+14 x^4+15 x^3+16 x^2+6 x+3$
- $y^2=14 x^6+11 x^5+8 x^4+3 x^3+2 x^2+13 x+10$
- $y^2=7 x^6+10 x^5+7 x^4+11 x^2+15 x+7$
- $y^2=7 x^6+11 x^5+11 x^4+x^3+14 x^2+7 x+7$
- $y^2=7 x^6+4 x^5+12 x^4+14 x^3+13 x^2+15 x+6$
- $y^2=6 x^6+13 x^5+5 x^4+15 x^3+7 x^2+x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{6}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{14})\). |
| The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.mmc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
- Endomorphism algebra over $\F_{17^{2}}$
The base change of $A$ to $\F_{17^{2}}$ is the simple isogeny class 2.289.w_hn and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{14})\). - Endomorphism algebra over $\F_{17^{3}}$
The base change of $A$ to $\F_{17^{3}}$ is the simple isogeny class 2.4913.a_mmc and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{14})\).
Base change
This is a primitive isogeny class.