Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 17 x^{2} )^{2}$ |
| $1 - 6 x + 43 x^{2} - 102 x^{3} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.381477984739$, $\pm0.381477984739$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $225$ | $99225$ | $25401600$ | $6968075625$ | $2009518880625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $340$ | $5166$ | $83428$ | $1415292$ | $24125470$ | $410379996$ | $6976087108$ | $118588163022$ | $2015989155700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=2 x^6+4 x^5+3 x^4+11 x^3+11 x^2+16 x+1$
- $y^2=9 x^6+4 x^5+2 x^4+13 x^3+x^2+x+16$
- $y^2=14 x^6+11 x^5+16 x^3+10 x+7$
- $y^2=10 x^6+16 x^5+9 x^4+10 x^3+16 x^2+13 x+5$
- $y^2=10 x^6+7 x^5+5 x^4+16 x^3+10 x^2+11 x+12$
- $y^2=11 x^6+4 x^5+4 x^4+12 x^3+2 x^2+x+12$
- $y^2=10 x^6+10 x^5+15 x^4+16 x^3+9 x^2+7 x+11$
- $y^2=12 x^6+9 x^5+2 x^4+x^3+15 x^2+9 x+5$
- $y^2=2 x^6+15 x^4+4 x^3+8 x^2+8$
- $y^2=12 x^6+14 x^5+11 x^4+9 x^3+14 x^2+12 x+10$
- $y^2=3 x^6+14 x^5+8 x^4+10 x^3+4 x^2+12 x+11$
- $y^2=5 x^6+12 x^5+8 x^4+9 x^3+2 x^2+5 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.