Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(29584\)\(\medspace = 2^{4} \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.118336.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 6.2.118336.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 3x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 12 + \left(11 a + 3\right)\cdot 19 + \left(5 a + 18\right)\cdot 19^{2} + \left(4 a + 12\right)\cdot 19^{3} + 6 a\cdot 19^{4} + \left(a + 17\right)\cdot 19^{5} + \left(6 a + 8\right)\cdot 19^{6} + \left(9 a + 2\right)\cdot 19^{7} + \left(a + 8\right)\cdot 19^{8} +O(19^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 6 + \left(7 a + 2\right)\cdot 19 + \left(13 a + 12\right)\cdot 19^{2} + \left(14 a + 11\right)\cdot 19^{3} + \left(12 a + 2\right)\cdot 19^{4} + \left(17 a + 12\right)\cdot 19^{5} + \left(12 a + 13\right)\cdot 19^{6} + \left(9 a + 5\right)\cdot 19^{7} + 17 a\cdot 19^{8} +O(19^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 4\cdot 19^{2} + 9\cdot 19^{3} + 16\cdot 19^{4} + 11\cdot 19^{5} + 19^{6} + 17\cdot 19^{7} + 18\cdot 19^{8} +O(19^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 6\cdot 19 + 13\cdot 19^{2} + 4\cdot 19^{3} + 7\cdot 19^{4} + 18\cdot 19^{6} + 14\cdot 19^{7} + 9\cdot 19^{8} +O(19^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 8 + \left(17 a + 1\right)\cdot 19 + \left(9 a + 18\right)\cdot 19^{2} + \left(6 a + 10\right)\cdot 19^{3} + \left(4 a + 6\right)\cdot 19^{4} + \left(4 a + 17\right)\cdot 19^{5} + 17 a\cdot 19^{6} + \left(2 a + 6\right)\cdot 19^{7} + \left(6 a + 8\right)\cdot 19^{8} +O(19^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 4 + \left(a + 4\right)\cdot 19 + \left(9 a + 10\right)\cdot 19^{2} + \left(12 a + 7\right)\cdot 19^{3} + \left(14 a + 4\right)\cdot 19^{4} + \left(14 a + 17\right)\cdot 19^{5} + \left(a + 13\right)\cdot 19^{6} + \left(16 a + 10\right)\cdot 19^{7} + \left(12 a + 11\right)\cdot 19^{8} +O(19^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)(3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.