Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 + 4 x + 17 x^{2} )^{3}$ |
| $1 + 12 x + 99 x^{2} + 472 x^{3} + 1683 x^{4} + 3468 x^{5} + 4913 x^{6}$ | |
| Frobenius angles: | $\pm0.661206336803$, $\pm0.661206336803$, $\pm0.661206336803$ |
| Angle rank: | $1$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 11$ |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10648$ | $29218112$ | $108804596824$ | $587975001112576$ | $2870686485614633048$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $344$ | $4494$ | $84284$ | $1423950$ | $24108248$ | $410386398$ | $6976065020$ | $118585834878$ | $2015996838104$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=3 x^8+11 x^7+12 x^6+3 x^5+14 x^4+6 x^3+9 x^2+4 x+2$
- $y^2=x^8+9 x^7+10 x^6+5 x^5+3 x^4+2 x^3+x^2+11 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.e 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This is a primitive isogeny class.