Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 4 x + 51 x^{2} + 132 x^{3} + 867 x^{4} + 1156 x^{5} + 4913 x^{6}$ |
| Frobenius angles: | $\pm0.465065257464$, $\pm0.546249397257$, $\pm0.648510336880$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.332130352.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7124$ | $32342960$ | $114986496404$ | $576804348640000$ | $2866588151358111284$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $376$ | $4762$ | $82684$ | $1421922$ | $24142408$ | $410337390$ | $6975723132$ | $118587366622$ | $2015995395816$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=3 x^8+3 x^7+14 x^6+12 x^5+4 x^4+14 x^3+8 x^2+13 x+9$
- $y^2=x^8+8 x^7+15 x^6+12 x^5+14 x^4+9 x^3+8 x^2+6 x+4$
- $y^2=3 x^8+10 x^7+11 x^6+2 x^5+2 x^4+5 x^3+15 x^2+5 x+15$
- $y^2=3 x^8+4 x^7+2 x^6+16 x^5+5 x^4+8 x^3+4 x^2+11 x+8$
- $y^2=x^8+3 x^7+14 x^6+10 x^5+2 x^4+4 x^3+5 x^2+x$
- $y^2=3 x^8+12 x^7+4 x^6+6 x^5+13 x^4+13 x^3+10 x^2+9$
- $y^2=3 x^8+5 x^7+9 x^6+8 x^4+x^3+14 x+1$
- $y^2=3 x^8+12 x^7+12 x^6+7 x^5+16 x^4+11 x^3+9 x^2+14 x+2$
- $y^2=3 x^8+16 x^7+6 x^6+x^5+x^4+6 x^3+2 x^2+6 x+13$
- $y^2=3 x^8+7 x^6+9 x^5+3 x^4+4 x^3+12 x^2+9 x+12$
- $y^2=x^8+13 x^7+3 x^6+14 x^5+6 x^4+12 x^3+11 x^2+x+5$
- $y^2=x^8+4 x^7+9 x^6+11 x^5+8 x^4+2 x^3+12 x^2+7$
- $y^2=3 x^8+7 x^5+15 x^4+4 x^3+11 x^2+4$
- $y^2=x^8+3 x^7+6 x^6+15 x^5+8 x^4+11 x^3+5 x+15$
- $y^2=3 x^8+11 x^7+10 x^6+2 x^5+5 x^4+14 x^3+8 x^2+x+11$
- $y^2=x^8+7 x^7+2 x^6+8 x^5+16 x^4+x^2+6 x+2$
- $y^2=x^8+6 x^7+15 x^6+6 x^5+15 x^4+x^3+3 x^2+2 x+10$
- $y^2=x^8+6 x^7+13 x^6+14 x^5+9 x^4+15 x^3+8 x^2+5 x+15$
- $y^2=x^8+11 x^7+7 x^6+7 x^5+9 x^4+6 x^3+7 x^2+9 x+4$
- $y^2=x^8+3 x^7+15 x^6+2 x^5+12 x^4+12 x^3+13 x^2+7 x+6$
- $y^2=x^8+15 x^7+14 x^6+5 x^5+x^4+11 x^3+7 x^2+5 x+10$
- $y^2=3 x^8+4 x^7+7 x^6+7 x^5+12 x^4+2 x^3+13 x+13$
- $y^2=3 x^8+3 x^7+6 x^6+12 x^5+9 x^4+9 x^3+5 x^2+10 x+7$
- $y^2=3 x^8+7 x^6+10 x^5+9 x^4+8 x^3+13 x^2+3 x+16$
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 6.0.332130352.1. |
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_bz_afc | $2$ | (not in LMFDB) |