Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x + 45 x^{2} - 37 x^{3} + 765 x^{4} - 289 x^{5} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.373902253579$, $\pm0.522932437026$, $\pm0.561697632134$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.26414663391.1 |
| Galois group: | $S_4\times C_2$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5397$ | $32657247$ | $119158943589$ | $575971717868631$ | $2863670637125558757$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $379$ | $4937$ | $82563$ | $1420477$ | $24143299$ | $410294104$ | $6975780995$ | $118589253989$ | $2015993076739$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=3 x^7+5 x^5+13 x^4+15 x^3+8 x^2+10 x+4$
- $y^2=x^7+8 x^5+3 x^4+11 x^3+2 x^2+14 x+9$
- $y^2=3 x^7+13 x^5+3 x^4+14 x^2+16 x+6$
- $y^2=x^7+10 x^5+9 x^4+14 x^3+3 x^2+4 x+3$
- $y^2=x^7+10 x^5+3 x^4+3 x^3+10 x^2+12 x+1$
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 6.0.26414663391.1. |
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.b_bt_bl | $2$ | (not in LMFDB) |