Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(805\)\(\medspace = 5 \cdot 7 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.18515.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.805.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.28175.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} - x^{3} + 2x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 9 + \left(8 a + 11\right)\cdot 17 + 13\cdot 17^{2} + \left(4 a + 16\right)\cdot 17^{3} + \left(11 a + 12\right)\cdot 17^{4} + \left(16 a + 6\right)\cdot 17^{5} + \left(5 a + 1\right)\cdot 17^{6} + \left(7 a + 7\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 13\cdot 17 + 3\cdot 17^{2} + 15\cdot 17^{3} + 5\cdot 17^{5} + 12\cdot 17^{6} + 10\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 13 + \left(15 a + 4\right)\cdot 17 + a\cdot 17^{2} + \left(15 a + 13\right)\cdot 17^{3} + \left(12 a + 13\right)\cdot 17^{4} + \left(5 a + 7\right)\cdot 17^{5} + \left(5 a + 2\right)\cdot 17^{6} + \left(8 a + 9\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 16 + \left(8 a + 12\right)\cdot 17 + \left(16 a + 5\right)\cdot 17^{2} + \left(12 a + 3\right)\cdot 17^{3} + \left(5 a + 3\right)\cdot 17^{4} + 12\cdot 17^{5} + \left(11 a + 7\right)\cdot 17^{6} + \left(9 a + 8\right)\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 16\cdot 17 + 6\cdot 17^{2} + 10\cdot 17^{3} + 8\cdot 17^{4} + 17^{5} + 8\cdot 17^{6} + 3\cdot 17^{7} +O(17^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 8 + \left(a + 8\right)\cdot 17 + \left(15 a + 3\right)\cdot 17^{2} + \left(a + 9\right)\cdot 17^{3} + \left(4 a + 11\right)\cdot 17^{4} + 11 a\cdot 17^{5} + \left(11 a + 2\right)\cdot 17^{6} + \left(8 a + 12\right)\cdot 17^{7} +O(17^{8})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.