Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 8 x^{2} + 85 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.374029230508$, $\pm0.959304102825$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-43})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $388$ | $80704$ | $25441936$ | $6934410496$ | $2015602095268$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $281$ | $5174$ | $83025$ | $1419583$ | $24123422$ | $410378719$ | $6975837409$ | $118588438358$ | $2015991136361$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=2 x^6+4 x^5+4 x^4+7 x^3+2 x^2+11 x+14$
- $y^2=10 x^6+3 x^5+x^4+6 x^3+4 x^2+11 x+3$
- $y^2=x^6+11 x^5+16 x^4+13 x^3+15 x^2+10$
- $y^2=6 x^6+12 x^5+12 x^4+12 x^3+3 x^2+11 x+16$
- $y^2=12 x^6+6 x^5+15 x^3+16 x^2+9 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{3}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-43})\). |
| The base change of $A$ to $\F_{17^{3}}$ is 1.4913.fa 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.