Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(2975\)\(\medspace = 5^{2} \cdot 7 \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.520625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.119.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.50575.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} - x^{3} + x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 44 + \left(9 a + 26\right)\cdot 47 + \left(a + 18\right)\cdot 47^{2} + \left(30 a + 11\right)\cdot 47^{3} + \left(13 a + 8\right)\cdot 47^{4} + \left(10 a + 44\right)\cdot 47^{5} + \left(30 a + 41\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 + 9\cdot 47 + 46\cdot 47^{2} + 45\cdot 47^{3} + 16\cdot 47^{4} + 46\cdot 47^{5} + 38\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 9 + \left(37 a + 39\right)\cdot 47 + \left(45 a + 11\right)\cdot 47^{2} + \left(16 a + 23\right)\cdot 47^{3} + \left(33 a + 5\right)\cdot 47^{4} + \left(36 a + 4\right)\cdot 47^{5} + \left(16 a + 45\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 12 + \left(9 a + 10\right)\cdot 47 + \left(23 a + 12\right)\cdot 47^{2} + 24\cdot 47^{3} + \left(44 a + 13\right)\cdot 47^{4} + \left(a + 36\right)\cdot 47^{5} + \left(45 a + 24\right)\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 47 + 3\cdot 47^{2} + 34\cdot 47^{3} + 42\cdot 47^{4} + 13\cdot 47^{5} + 18\cdot 47^{6} +O(47^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 11 + \left(37 a + 6\right)\cdot 47 + \left(23 a + 2\right)\cdot 47^{2} + \left(46 a + 2\right)\cdot 47^{3} + \left(2 a + 7\right)\cdot 47^{4} + \left(45 a + 43\right)\cdot 47^{5} + \left(a + 18\right)\cdot 47^{6} +O(47^{7})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.