Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(5952\)\(\medspace = 2^{6} \cdot 3 \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.184512.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.372.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.17856.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{4} + 4x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 + 27\cdot 47 + 5\cdot 47^{2} + 35\cdot 47^{3} + 43\cdot 47^{4} + 33\cdot 47^{5} + 7\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 a + 10 + \left(35 a + 6\right)\cdot 47 + \left(20 a + 44\right)\cdot 47^{2} + \left(29 a + 27\right)\cdot 47^{3} + \left(8 a + 29\right)\cdot 47^{4} + \left(11 a + 16\right)\cdot 47^{5} + \left(21 a + 31\right)\cdot 47^{6} + \left(30 a + 3\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 28 + \left(7 a + 25\right)\cdot 47 + \left(17 a + 33\right)\cdot 47^{2} + \left(40 a + 38\right)\cdot 47^{3} + \left(33 a + 9\right)\cdot 47^{4} + \left(a + 15\right)\cdot 47^{5} + \left(14 a + 10\right)\cdot 47^{6} + \left(2 a + 28\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 19\cdot 47 + 41\cdot 47^{2} + 11\cdot 47^{3} + 3\cdot 47^{4} + 13\cdot 47^{5} + 46\cdot 47^{6} + 39\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 37 + \left(11 a + 40\right)\cdot 47 + \left(26 a + 2\right)\cdot 47^{2} + \left(17 a + 19\right)\cdot 47^{3} + \left(38 a + 17\right)\cdot 47^{4} + \left(35 a + 30\right)\cdot 47^{5} + \left(25 a + 15\right)\cdot 47^{6} + \left(16 a + 43\right)\cdot 47^{7} +O(47^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 a + 19 + \left(39 a + 21\right)\cdot 47 + \left(29 a + 13\right)\cdot 47^{2} + \left(6 a + 8\right)\cdot 47^{3} + \left(13 a + 37\right)\cdot 47^{4} + \left(45 a + 31\right)\cdot 47^{5} + \left(32 a + 36\right)\cdot 47^{6} + \left(44 a + 18\right)\cdot 47^{7} +O(47^{8})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ | |
| $3$ | $2$ | $(3,6)$ | $1$ | |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ | |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ | ✓ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ | |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | |
| $6$ | $4$ | $(1,3,4,6)$ | $1$ | |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $-1$ | |
| $8$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |