Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 17 x^{2} )( 1 - 2 x + 17 x^{2} )$ |
| $1 - 7 x + 44 x^{2} - 119 x^{3} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.292637436158$, $\pm0.422020869623$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $7$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $208$ | $95680$ | $25260352$ | $6990380800$ | $2013203262928$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $329$ | $5138$ | $83697$ | $1417891$ | $24131486$ | $410338723$ | $6975740833$ | $118587602546$ | $2015994477689$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=14 x^6+12 x^5+8 x^4+9 x^3+3 x^2+6 x+4$
- $y^2=6 x^6+3 x^5+6 x^4+6 x^3+16 x^2+7 x$
- $y^2=5 x^6+8 x^4+x^3+x^2+5 x+14$
- $y^2=16 x^6+12 x^5+6 x^4+13 x^3+13 x^2+8 x+7$
- $y^2=5 x^6+10 x^5+16 x^4+2 x^3+10 x^2+3 x$
- $y^2=6 x^6+6 x^5+8 x^4+12 x^3+6 x^2+10 x+3$
- $y^2=3 x^6+2 x^5+15 x^4+14 x^3+12 x^2+4 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.af $\times$ 1.17.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.