Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1265\)\(\medspace = 5 \cdot 11 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.29095.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.1265.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.69575.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} - x^{3} + 3x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 a + 35 + \left(5 a + 22\right)\cdot 43 + \left(41 a + 31\right)\cdot 43^{2} + \left(37 a + 22\right)\cdot 43^{3} + \left(19 a + 31\right)\cdot 43^{4} + \left(25 a + 31\right)\cdot 43^{5} + \left(31 a + 25\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 28\cdot 43 + 36\cdot 43^{2} + 25\cdot 43^{3} + 28\cdot 43^{4} + 33\cdot 43^{5} +O(43^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 17 + \left(37 a + 3\right)\cdot 43 + \left(a + 24\right)\cdot 43^{2} + \left(5 a + 19\right)\cdot 43^{3} + \left(23 a + 13\right)\cdot 43^{4} + \left(17 a + 37\right)\cdot 43^{5} + \left(11 a + 31\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 29 + \left(11 a + 27\right)\cdot 43 + \left(20 a + 7\right)\cdot 43^{2} + \left(a + 32\right)\cdot 43^{3} + \left(29 a + 15\right)\cdot 43^{4} + \left(16 a + 3\right)\cdot 43^{5} + \left(19 a + 34\right)\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 + 40\cdot 43 + 12\cdot 43^{2} + 15\cdot 43^{3} + 39\cdot 43^{4} + 31\cdot 43^{5} + 42\cdot 43^{6} +O(43^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 19 + \left(31 a + 6\right)\cdot 43 + \left(22 a + 16\right)\cdot 43^{2} + \left(41 a + 13\right)\cdot 43^{3} + 13 a\cdot 43^{4} + \left(26 a + 34\right)\cdot 43^{5} + \left(23 a + 36\right)\cdot 43^{6} +O(43^{7})\)
|
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.