Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 4 x + 47 x^{2} + 116 x^{3} + 799 x^{4} + 1156 x^{5} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.415785220772$, $\pm0.603361508136$, $\pm0.640612955973$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.294367856.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7036$ | $31549424$ | $114945225244$ | $580368564971264$ | $2866421209129110716$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $368$ | $4762$ | $83196$ | $1421842$ | $24125120$ | $410344670$ | $6976141436$ | $118587041326$ | $2015987753568$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=3 x^8+8 x^7+x^6+9 x^4+7 x^3+5 x^2+14 x+13$
- $y^2=3 x^8+14 x^7+15 x^6+2 x^5+13 x^4+11 x^3+4 x^2+9 x+1$
- $y^2=x^7+13 x^6+14 x^5+x^4+16 x^3+5 x^2+8 x+12$
- $y^2=3 x^7+9 x^6+7 x^5+7 x^4+14 x^3+15 x^2+9 x+9$
- $y^2=x^8+15 x^7+16 x^6+7 x^5+15 x^4+8 x^3+16 x^2+5 x+9$
- $y^2=x^8+3 x^7+6 x^6+3 x^5+13 x^4+7 x^3+7 x^2+8$
- $y^2=3 x^8+2 x^7+13 x^6+2 x^5+x^4+x^3+5 x+6$
- $y^2=3 x^8+16 x^7+3 x^6+7 x^5+5 x^4+10 x^3+8 x^2+11 x+6$
- $y^2=x^8+9 x^6+11 x^5+16 x^4+16 x^3+13 x^2+2 x+2$
- $y^2=x^8+12 x^7+8 x^6+10 x^5+14 x^4+5 x^3+16 x^2+3 x+15$
- $y^2=x^8+3 x^7+13 x^6+x^5+4 x^4+4 x^3+2 x^2+2 x+15$
- $y^2=3 x^8+13 x^7+3 x^6+6 x^4+10 x^3+4 x^2+7 x+15$
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 6.0.294367856.1. |
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.ae_bv_aem | $2$ | (not in LMFDB) |