Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(856\)\(\medspace = 2^{3} \cdot 107 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.91592.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.856.2t1.b.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.6848.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 36 + \left(46 a + 32\right)\cdot 47 + \left(32 a + 12\right)\cdot 47^{2} + \left(5 a + 4\right)\cdot 47^{3} + \left(16 a + 27\right)\cdot 47^{4} + \left(46 a + 30\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 37\cdot 47 + 32\cdot 47^{2} + 29\cdot 47^{3} + 46\cdot 47^{4} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 31\cdot 47 + 33\cdot 47^{2} + 8\cdot 47^{3} + 25\cdot 47^{4} + 16\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 a + 7 + 22\cdot 47 + \left(14 a + 32\right)\cdot 47^{2} + \left(41 a + 29\right)\cdot 47^{3} + \left(30 a + 6\right)\cdot 47^{4} + 13\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 8 + \left(2 a + 13\right)\cdot 47 + \left(11 a + 28\right)\cdot 47^{2} + \left(32 a + 7\right)\cdot 47^{3} + \left(46 a + 34\right)\cdot 47^{4} + \left(40 a + 45\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 36 + \left(44 a + 3\right)\cdot 47 + \left(35 a + 1\right)\cdot 47^{2} + \left(14 a + 14\right)\cdot 47^{3} + 47^{4} + \left(6 a + 34\right)\cdot 47^{5} +O(47^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,6,4,5)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,6,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.