Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 3 x + 47 x^{2} + 98 x^{3} + 799 x^{4} + 867 x^{5} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.442195159688$, $\pm0.527472717754$, $\pm0.651507526360$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.5954612992.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6728$ | $32240576$ | $116159754272$ | $577157379472384$ | $2863962583905140168$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $375$ | $4812$ | $82735$ | $1420621$ | $24141228$ | $410362029$ | $6975740319$ | $118587045516$ | $2015995191655$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 21 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+13 x^7+8 x^6+16 x^5+6 x^4+2 x^3+16 x^2+16 x$
- $y^2=x^8+15 x^7+16 x^6+6 x^5+10 x^4+6 x^3+8 x^2+16 x$
- $y^2=x^8+10 x^7+12 x^6+7 x^4+12 x^3+2 x^2+15 x$
- $y^2=3 x^8+8 x^7+6 x^6+2 x^5+11 x^4+7 x^3+2 x^2+x$
- $y^2=3 x^8+5 x^7+8 x^6+14 x^5+3 x^4+x^3+2 x^2+4 x$
- $y^2=3 x^8+9 x^7+13 x^6+16 x^5+14 x^4+4 x^3+8$
- $y^2=x^8+7 x^7+2 x^6+15 x^5+16 x^4+7 x^3+14 x^2+6 x$
- $y^2=x^8+x^7+16 x^6+11 x^5+12 x^4+3 x^3+x^2+7 x+11$
- $y^2=3 x^8+4 x^7+11 x^6+11 x^5+12 x^4+5 x^3+8 x+4$
- $y^2=x^7+13 x^6+15 x^5+3 x^4+9 x^3+6 x^2+4 x+15$
- $y^2=3 x^7+x^6+2 x^5+4 x^4+11 x^3+4 x^2+15 x+12$
- $y^2=3 x^7+2 x^6+2 x^5+2 x^4+15 x^3+x^2+8 x+8$
- $y^2=x^7+5 x^6+5 x^5+9 x^4+16 x^3+12 x^2+10 x+2$
- $y^2=x^7+16 x^6+5 x^5+x^4+7 x^3+15 x^2+15 x+3$
- $y^2=3 x^8+8 x^7+15 x^5+6 x^4+13 x^3+7 x^2+5 x+13$
- $y^2=3 x^8+9 x^7+7 x^6+4 x^5+6 x^4+14 x^3+6 x^2+6 x+15$
- $y^2=3 x^8+2 x^7+6 x^6+9 x^4+x^3+14$
- $y^2=x^8+9 x^7+16 x^5+14 x^4+4 x^3+3 x^2+4 x+4$
- $y^2=3 x^8+3 x^7+x^6+11 x^5+14 x^4+3 x^3+13 x^2+5 x+13$
- $y^2=x^8+11 x^7+12 x^6+10 x^5+3 x^4+5 x^3+10 x^2+6 x+16$
- $y^2=x^8+15 x^7+4 x^6+6 x^5+5 x^4+7 x^3+12 x^2+4 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is 6.0.5954612992.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.17.ad_bv_adu | $2$ | (not in LMFDB) |