Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(219501\)\(\medspace = 3^{2} \cdot 29^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.19096587.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.29.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.73167.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 8x^{4} - 6x^{3} + 18x^{2} + 54x + 45 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 15 + \left(5 a + 9\right)\cdot 19 + \left(16 a + 10\right)\cdot 19^{2} + \left(17 a + 12\right)\cdot 19^{3} + \left(14 a + 17\right)\cdot 19^{4} + \left(4 a + 5\right)\cdot 19^{5} + \left(14 a + 11\right)\cdot 19^{6} + 2 a\cdot 19^{7} + \left(2 a + 3\right)\cdot 19^{8} + 9 a\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 16\cdot 19 + 2\cdot 19^{2} + 4\cdot 19^{4} + 8\cdot 19^{5} + 16\cdot 19^{6} + 11\cdot 19^{7} + 15\cdot 19^{8} + 18\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 4 + \left(13 a + 7\right)\cdot 19 + \left(2 a + 2\right)\cdot 19^{2} + \left(a + 14\right)\cdot 19^{3} + \left(4 a + 14\right)\cdot 19^{4} + \left(14 a + 14\right)\cdot 19^{5} + \left(4 a + 1\right)\cdot 19^{6} + \left(16 a + 8\right)\cdot 19^{7} + \left(16 a + 2\right)\cdot 19^{8} + \left(9 a + 7\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 11\cdot 19 + 13\cdot 19^{2} + 3\cdot 19^{3} + 11\cdot 19^{4} + 7\cdot 19^{5} + 18\cdot 19^{7} + 12\cdot 19^{8} + 10\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 18 + \left(11 a + 7\right)\cdot 19 + \left(16 a + 11\right)\cdot 19^{2} + \left(6 a + 8\right)\cdot 19^{3} + \left(4 a + 15\right)\cdot 19^{4} + \left(10 a + 16\right)\cdot 19^{5} + \left(3 a + 16\right)\cdot 19^{6} + \left(11 a + 14\right)\cdot 19^{7} + \left(8 a + 12\right)\cdot 19^{8} + \left(12 a + 17\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 15 + \left(7 a + 3\right)\cdot 19 + \left(2 a + 16\right)\cdot 19^{2} + \left(12 a + 17\right)\cdot 19^{3} + \left(14 a + 12\right)\cdot 19^{4} + \left(8 a + 3\right)\cdot 19^{5} + \left(15 a + 10\right)\cdot 19^{6} + \left(7 a + 3\right)\cdot 19^{7} + \left(10 a + 10\right)\cdot 19^{8} + \left(6 a + 2\right)\cdot 19^{9} +O(19^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(2,4)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(2,3,4,5)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,3,4,5)$ | $-1$ |
| $8$ | $6$ | $(1,3,4,6,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.