Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(908087\)\(\medspace = 47 \cdot 139^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.126224093.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.47.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.307051.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 18x^{4} + 41x^{3} + 54x^{2} - 75x - 82 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 + 7\cdot 23 + 19\cdot 23^{2} + 7\cdot 23^{3} + 8\cdot 23^{4} + 12\cdot 23^{5} + 4\cdot 23^{6} + 4\cdot 23^{7} + 16\cdot 23^{8} + 14\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 1 + \left(21 a + 3\right)\cdot 23 + \left(20 a + 1\right)\cdot 23^{2} + \left(16 a + 9\right)\cdot 23^{3} + \left(6 a + 11\right)\cdot 23^{4} + \left(9 a + 15\right)\cdot 23^{5} + \left(4 a + 6\right)\cdot 23^{6} + \left(15 a + 19\right)\cdot 23^{7} + \left(3 a + 7\right)\cdot 23^{8} + \left(9 a + 5\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 12 + \left(a + 6\right)\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(6 a + 21\right)\cdot 23^{3} + \left(16 a + 7\right)\cdot 23^{4} + \left(13 a + 4\right)\cdot 23^{5} + \left(18 a + 6\right)\cdot 23^{6} + \left(7 a + 22\right)\cdot 23^{7} + \left(19 a + 22\right)\cdot 23^{8} + \left(13 a + 19\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + \left(a + 20\right)\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(6 a + 13\right)\cdot 23^{3} + \left(16 a + 11\right)\cdot 23^{4} + \left(13 a + 7\right)\cdot 23^{5} + \left(18 a + 16\right)\cdot 23^{6} + \left(7 a + 3\right)\cdot 23^{7} + \left(19 a + 15\right)\cdot 23^{8} + \left(13 a + 17\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 12 + \left(21 a + 16\right)\cdot 23 + \left(20 a + 1\right)\cdot 23^{2} + \left(16 a + 1\right)\cdot 23^{3} + \left(6 a + 15\right)\cdot 23^{4} + \left(9 a + 18\right)\cdot 23^{5} + \left(4 a + 16\right)\cdot 23^{6} + 15 a\cdot 23^{7} + 3 a\cdot 23^{8} + \left(9 a + 3\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 15\cdot 23 + 3\cdot 23^{2} + 15\cdot 23^{3} + 14\cdot 23^{4} + 10\cdot 23^{5} + 18\cdot 23^{6} + 18\cdot 23^{7} + 6\cdot 23^{8} + 8\cdot 23^{9} +O(23^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)(3,5)$ | $-1$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,6,5)$ | $1$ |
| $6$ | $4$ | $(1,4,6,2)(3,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.