Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(7098343\)\(\medspace = 7 \cdot 19^{2} \cdot 53^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.7098343.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.49343.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + 5x^{3} + x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 23 + \left(14 a + 7\right)\cdot 29 + \left(23 a + 27\right)\cdot 29^{2} + \left(25 a + 3\right)\cdot 29^{3} + \left(21 a + 27\right)\cdot 29^{4} + \left(8 a + 18\right)\cdot 29^{5} + \left(16 a + 5\right)\cdot 29^{6} + \left(21 a + 22\right)\cdot 29^{7} + \left(15 a + 7\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 23\cdot 29 + 12\cdot 29^{2} + 7\cdot 29^{3} + 3\cdot 29^{4} + 4\cdot 29^{5} + 27\cdot 29^{6} + 23\cdot 29^{7} + 4\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 11 + \left(11 a + 6\right)\cdot 29 + \left(24 a + 18\right)\cdot 29^{2} + \left(19 a + 5\right)\cdot 29^{3} + \left(20 a + 11\right)\cdot 29^{4} + \left(7 a + 6\right)\cdot 29^{5} + \left(7 a + 4\right)\cdot 29^{6} + \left(8 a + 24\right)\cdot 29^{7} + \left(23 a + 24\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 16 + \left(14 a + 6\right)\cdot 29 + \left(5 a + 14\right)\cdot 29^{2} + \left(3 a + 22\right)\cdot 29^{3} + \left(7 a + 23\right)\cdot 29^{4} + \left(20 a + 11\right)\cdot 29^{5} + \left(12 a + 20\right)\cdot 29^{6} + \left(7 a + 26\right)\cdot 29^{7} + \left(13 a + 6\right)\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 26\cdot 29 + 29^{2} + 25\cdot 29^{3} + 13\cdot 29^{4} + 21\cdot 29^{5} + 25\cdot 29^{6} + 18\cdot 29^{7} + 25\cdot 29^{8} +O(29^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 19 + \left(17 a + 16\right)\cdot 29 + \left(4 a + 12\right)\cdot 29^{2} + \left(9 a + 22\right)\cdot 29^{3} + \left(8 a + 7\right)\cdot 29^{4} + \left(21 a + 24\right)\cdot 29^{5} + \left(21 a + 3\right)\cdot 29^{6} + 20 a\cdot 29^{7} + \left(5 a + 17\right)\cdot 29^{8} +O(29^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.