Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 7 x + 42 x^{2} + 119 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.555810648148$, $\pm0.735613733058$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.706316.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $458$ | $94348$ | $23260904$ | $6986280704$ | $2016799483018$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $325$ | $4732$ | $83649$ | $1420425$ | $24139810$ | $410336665$ | $6975573249$ | $118588819804$ | $2015994625125$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=4 x^6+3 x^5+5 x^4+13 x^3+3 x^2+9 x+15$
- $y^2=12 x^6+11 x^5+4 x^3+9 x^2+2$
- $y^2=2 x^6+9 x^5+3 x^4+13 x^3+15 x^2+3 x+1$
- $y^2=4 x^6+2 x^5+16 x^4+2 x^3+9 x^2+4 x+14$
- $y^2=13 x^6+2 x^5+13 x^4+14 x^3+5 x^2+15 x+8$
- $y^2=x^6+11 x^5+4 x^4+8 x^3+5 x^2+14 x+15$
- $y^2=4 x^6+16 x^5+10 x^4+4 x^3+12 x^2+2 x+16$
- $y^2=x^6+x^5+7 x^4+15 x^3+14 x+14$
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 4.0.706316.1. |
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 |
|---|---|---|
| 2.17.ah_bq | $2$ | (not in LMFDB) |