Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 29 x^{2} - 17 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.390082729593$, $\pm0.569700715483$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1720341.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $301$ | $101437$ | $24309061$ | $6935957749$ | $2015656666576$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $347$ | $4949$ | $83043$ | $1419622$ | $24135347$ | $410325037$ | $6975942691$ | $118588487633$ | $2015989549982$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=8 x^5+16 x^4+16 x^3+3 x^2+x+12$
- $y^2=12 x^6+5 x^5+15 x^4+14 x^3+12 x^2+12 x+1$
- $y^2=14 x^6+5 x^5+7 x^4+8 x^3+15 x^2+2 x+4$
- $y^2=7 x^6+10 x^5+9 x^4+9 x^3+2 x^2+9 x+16$
- $y^2=14 x^5+11 x^4+4 x^3+8 x^2+16 x+10$
- $y^2=6 x^6+9 x^5+2 x^4+6 x^3+6 x^2+12 x+3$
- $y^2=6 x^6+6 x^5+12 x^4+11 x^3+13 x^2+9 x+12$
- $y^2=4 x^6+4 x^5+12 x^4+9 x^3+9 x^2+4 x+6$
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 4.0.1720341.1. |
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.b_bd | $2$ | (not in LMFDB) |