Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1003\)\(\medspace = 17 \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.59177.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.1003.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.17051.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - x^{3} + x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 18 + \left(16 a + 30\right)\cdot 43 + \left(2 a + 41\right)\cdot 43^{2} + \left(22 a + 38\right)\cdot 43^{3} + \left(5 a + 1\right)\cdot 43^{4} + 16\cdot 43^{5} +O(43^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 13 + \left(32 a + 3\right)\cdot 43 + \left(7 a + 15\right)\cdot 43^{2} + \left(36 a + 22\right)\cdot 43^{3} + \left(41 a + 28\right)\cdot 43^{4} + \left(30 a + 39\right)\cdot 43^{5} +O(43^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 a + 38 + \left(26 a + 26\right)\cdot 43 + \left(40 a + 27\right)\cdot 43^{2} + \left(20 a + 15\right)\cdot 43^{3} + \left(37 a + 28\right)\cdot 43^{4} + \left(42 a + 10\right)\cdot 43^{5} +O(43^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a + 37 + \left(10 a + 11\right)\cdot 43 + \left(35 a + 33\right)\cdot 43^{2} + \left(6 a + 7\right)\cdot 43^{3} + \left(a + 34\right)\cdot 43^{4} + \left(12 a + 28\right)\cdot 43^{5} +O(43^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 + 28\cdot 43 + 18\cdot 43^{2} + 29\cdot 43^{3} + 42\cdot 43^{4} + 22\cdot 43^{5} +O(43^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 28\cdot 43 + 35\cdot 43^{2} + 14\cdot 43^{3} + 36\cdot 43^{4} + 10\cdot 43^{5} +O(43^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.