Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 9 x^{2} - 98 x^{3} + 153 x^{4} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.142900649621$, $\pm0.615865403422$, $\pm0.683429652442$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.20783335104.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4978$ | $25756172$ | $111685297906$ | $585783643437616$ | $2871327830149209458$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $308$ | $4620$ | $83972$ | $1424268$ | $24131432$ | $410388066$ | $6976087580$ | $118587600138$ | $2015994497888$ |
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:
- $y^2=3 x^8+12 x^7+10 x^6+7 x^5+3 x^4+6 x^3+14 x^2+9$
- $y^2=x^8+2 x^7+9 x^6+16 x^5+6 x^4+x^3+13 x^2+9$
- $y^2=x^8+16 x^7+6 x^5+15 x^4+11 x^3+16 x^2+9 x+8$
- $y^2=3 x^8+15 x^7+12 x^6+15 x^5+7 x^4+2 x^3+10 x^2+5$
- $y^2=x^8+14 x^7+x^6+7 x^5+8 x^4+4 x^3+8 x^2+11 x+3$
- $y^2=3 x^8+2 x^7+16 x^6+5 x^5+2 x^3+15 x^2+10 x+12$
- $y^2=x^8+15 x^7+6 x^6+14 x^5+6 x^4+8 x^2+3 x+2$
- $y^2=3 x^8+14 x^7+4 x^6+x^5+7 x^4+14 x^3+10 x^2+15 x+12$
- $y^2=3 x^8+9 x^7+5 x^5+14 x^4+9 x^3+9 x^2+10 x+10$
- $y^2=x^8+9 x^6+3 x^5+12 x^4+9 x^3+15 x^2+4 x+12$
- $y^2=x^8+4 x^6+8 x^5+15 x^4+2 x^3+7 x^2+5 x+15$
- $y^2=3 x^8+16 x^7+4 x^6+4 x^5+15 x^4+8 x^3+13 x^2+8 x+12$
- $y^2=x^8+3 x^7+x^6+13 x^5+10 x^4+x^3+12 x^2+4 x+8$
- $y^2=3 x^8+9 x^7+15 x^6+7 x^5+9 x^4+16 x^3+5 x^2+x+12$
- $y^2=3 x^8+14 x^7+15 x^6+8 x^5+10 x^4+8 x^3+13 x^2+10 x+6$
- $y^2=3 x^8+7 x^7+x^6+5 x^5+16 x^4+x^3+3 x^2+9 x+14$
- $y^2=3 x^8+14 x^7+14 x^6+9 x^5+12 x^4+12 x^2+6 x+11$
- $y^2=x^8+3 x^7+10 x^6+5 x^5+6 x^4+16 x^3+13 x^2+8$
- $y^2=3 x^8+2 x^7+10 x^6+2 x^5+8 x^3+2 x^2+x+8$
- $y^2=3 x^8+15 x^7+9 x^6+13 x^5+3 x^4+16 x^3+10 x^2+3 x+9$
- $y^2=x^8+4 x^7+7 x^6+11 x^5+14 x^4+6 x^3+12 x^2+6 x+5$
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 6.0.20783335104.1. |
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.a_j_du | $2$ | (not in LMFDB) |