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: | $\pm0.359387044027$, $\pm0.396638491864$, $\pm0.584214779228$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4484$ | $31549424$ | $122282881508$ | $580368564971264$ | $2858421755115509764$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $368$ | $5066$ | $83196$ | $1417874$ | $24125120$ | $410332678$ | $6976141436$ | $118588711670$ | $2015987753568$ |
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:
- $y^2=x^8+14 x^7+6 x^6+3 x^4+8 x^3+13 x^2+16 x+10$
- $y^2=x^8+16 x^7+5 x^6+12 x^5+10 x^4+15 x^3+7 x^2+3 x+6$
- $y^2=3 x^7+5 x^6+8 x^5+3 x^4+14 x^3+15 x^2+7 x+2$
- $y^2=x^7+3 x^6+8 x^5+8 x^4+16 x^3+5 x^2+3 x+3$
- $y^2=3 x^8+11 x^7+14 x^6+4 x^5+11 x^4+7 x^3+14 x^2+15 x+10$
- $y^2=3 x^8+9 x^7+x^6+9 x^5+5 x^4+4 x^3+4 x^2+7$
- $y^2=x^8+12 x^7+10 x^6+12 x^5+6 x^4+6 x^3+13 x+2$
- $y^2=x^8+11 x^7+x^6+8 x^5+13 x^4+9 x^3+14 x^2+15 x+2$
- $y^2=3 x^8+10 x^6+16 x^5+14 x^4+14 x^3+5 x^2+6 x+6$
- $y^2=3 x^8+2 x^7+7 x^6+13 x^5+8 x^4+15 x^3+14 x^2+9 x+11$
- $y^2=3 x^8+9 x^7+5 x^6+3 x^5+12 x^4+12 x^3+6 x^2+6 x+11$
- $y^2=x^8+10 x^7+x^6+2 x^4+9 x^3+7 x^2+8 x+5$
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 6.0.294367856.1. |
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.e_bv_em | $2$ | (not in LMFDB) |