Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(56603\)\(\medspace = 23^{2} \cdot 107 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.56603.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.107.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.263327.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} + x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 a + 34 + \left(14 a + 4\right)\cdot 37 + \left(27 a + 18\right)\cdot 37^{2} + \left(21 a + 16\right)\cdot 37^{3} + \left(16 a + 22\right)\cdot 37^{4} + \left(10 a + 6\right)\cdot 37^{5} + \left(12 a + 25\right)\cdot 37^{6} + \left(23 a + 35\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 8 + \left(17 a + 12\right)\cdot 37 + \left(31 a + 20\right)\cdot 37^{2} + \left(17 a + 2\right)\cdot 37^{3} + \left(a + 12\right)\cdot 37^{4} + \left(32 a + 22\right)\cdot 37^{5} + \left(36 a + 26\right)\cdot 37^{6} + \left(10 a + 14\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 20\cdot 37 + 11\cdot 37^{2} + 37^{3} + 12\cdot 37^{4} + 20\cdot 37^{5} + 20\cdot 37^{6} + 8\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 31 + \left(22 a + 36\right)\cdot 37 + \left(9 a + 1\right)\cdot 37^{2} + \left(15 a + 2\right)\cdot 37^{3} + \left(20 a + 30\right)\cdot 37^{4} + \left(26 a + 31\right)\cdot 37^{5} + \left(24 a + 26\right)\cdot 37^{6} + \left(13 a + 5\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 35\cdot 37 + 3\cdot 37^{2} + 9\cdot 37^{3} + 34\cdot 37^{4} + 28\cdot 37^{5} + 17\cdot 37^{6} + 24\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 a + 32 + 19 a\cdot 37 + \left(5 a + 18\right)\cdot 37^{2} + \left(19 a + 5\right)\cdot 37^{3} + 35 a\cdot 37^{4} + \left(4 a + 1\right)\cdot 37^{5} + 31\cdot 37^{6} + \left(26 a + 21\right)\cdot 37^{7} +O(37^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(4,6)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,3)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,2,4)$ | $-1$ |
| $6$ | $4$ | $(1,6,2,4)(3,5)$ | $1$ |
| $8$ | $6$ | $(1,6,5,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.