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: | $\pm0.618522015261$, $\pm0.618522015261$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $441$ | $99225$ | $22924944$ | $6968075625$ | $2022485023881$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $340$ | $4662$ | $83428$ | $1424424$ | $24125470$ | $410297352$ | $6976087108$ | $118587589974$ | $2015989155700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=12 x^6+7 x^5+x^4+15 x^3+15 x^2+11 x+6$
- $y^2=10 x^6+12 x^5+6 x^4+5 x^3+3 x^2+3 x+14$
- $y^2=8 x^6+16 x^5+14 x^3+13 x+4$
- $y^2=13 x^6+14 x^5+10 x^4+13 x^3+14 x^2+5 x+15$
- $y^2=9 x^6+8 x^5+13 x^4+11 x^3+9 x^2+15 x+4$
- $y^2=16 x^6+12 x^5+12 x^4+2 x^3+6 x^2+3 x+2$
- $y^2=9 x^6+9 x^5+5 x^4+11 x^3+3 x^2+8 x+15$
- $y^2=2 x^6+10 x^5+6 x^4+3 x^3+11 x^2+10 x+15$
- $y^2=12 x^6+5 x^4+7 x^3+14 x^2+14$
- $y^2=2 x^6+8 x^5+16 x^4+10 x^3+8 x^2+2 x+13$
- $y^2=x^6+16 x^5+14 x^4+9 x^3+7 x^2+4 x+15$
- $y^2=13 x^6+4 x^5+14 x^4+3 x^3+12 x^2+13 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.