Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 29 x^{2} + 85 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.466971795190$, $\pm0.751266277383$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.56725.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $409$ | $93661$ | $23872921$ | $6979711381$ | $2011881182464$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $323$ | $4859$ | $83571$ | $1416958$ | $24145787$ | $410393419$ | $6975477763$ | $118587826943$ | $2015995230878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=4 x^6+13 x^5+5 x^4+12 x^3+3 x^2+7 x+4$
- $y^2=x^6+13 x^5+5 x^4+4 x^3+4 x^2+5 x+14$
- $y^2=13 x^6+9 x^5+14 x^4+13 x^3+6 x+2$
- $y^2=14 x^6+7 x^5+16 x^4+16 x^3+3 x^2+7 x+1$
- $y^2=x^6+8 x^5+5 x^4+8 x^3+8 x^2+15$
- $y^2=6 x^6+12 x^4+2 x^3+9 x+1$
- $y^2=4 x^6+7 x^5+3 x^4+11 x^3+9 x^2+13$
- $y^2=8 x^6+5 x^5+9 x^3+16 x^2+4 x+8$
- $y^2=8 x^5+15 x^4+14 x^3+5 x^2+15 x+3$
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 4.0.56725.1. |
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.af_bd | $2$ | (not in LMFDB) |