Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 17 x^{2} )^{2}$ |
| $1 - 8 x + 50 x^{2} - 136 x^{3} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.338793663197$, $\pm0.338793663197$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 7$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $196$ | $94864$ | $25542916$ | $7018418176$ | $2012125228036$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $326$ | $5194$ | $84030$ | $1417130$ | $24118022$ | $410306858$ | $6975962494$ | $118589237578$ | $2015995858886$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=6 x^6+6 x^5+8 x^4+10 x^3+8 x^2+6 x+6$
- $y^2=3 x^6+14 x^5+15 x^4+5 x^3+2 x^2+x+8$
- $y^2=12 x^6+15 x^4+15 x^2+12$
- $y^2=2 x^6+13 x^5+11 x^4+x^3+11 x^2+5 x+5$
- $y^2=14 x^6+8 x^5+10 x^4+3 x^3+10 x^2+8 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.