Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(91375\)\(\medspace = 5^{3} \cdot 17 \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.66795125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.3655.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.731.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 7x^{4} - 11x^{2} - 13x - 14 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 20 + \left(9 a + 27\right)\cdot 29 + \left(12 a + 19\right)\cdot 29^{2} + \left(18 a + 13\right)\cdot 29^{3} + \left(28 a + 22\right)\cdot 29^{4} + \left(22 a + 3\right)\cdot 29^{5} + \left(3 a + 28\right)\cdot 29^{6} + \left(22 a + 3\right)\cdot 29^{7} + \left(24 a + 15\right)\cdot 29^{8} + \left(8 a + 16\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 11 + \left(10 a + 28\right)\cdot 29 + \left(7 a + 22\right)\cdot 29^{2} + \left(24 a + 27\right)\cdot 29^{3} + \left(3 a + 24\right)\cdot 29^{4} + \left(18 a + 1\right)\cdot 29^{5} + \left(5 a + 27\right)\cdot 29^{6} + \left(5 a + 26\right)\cdot 29^{7} + \left(10 a + 25\right)\cdot 29^{8} + \left(25 a + 10\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 13\cdot 29 + 26\cdot 29^{2} + 20\cdot 29^{3} + 14\cdot 29^{5} + 4\cdot 29^{7} + 6\cdot 29^{8} + 3\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 8 + \left(18 a + 11\right)\cdot 29 + \left(21 a + 20\right)\cdot 29^{2} + \left(4 a + 25\right)\cdot 29^{3} + \left(25 a + 19\right)\cdot 29^{4} + \left(10 a + 1\right)\cdot 29^{5} + \left(23 a + 8\right)\cdot 29^{6} + \left(23 a + 18\right)\cdot 29^{7} + \left(18 a + 13\right)\cdot 29^{8} + \left(3 a + 11\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 3 + \left(19 a + 3\right)\cdot 29 + \left(16 a + 14\right)\cdot 29^{2} + \left(10 a + 6\right)\cdot 29^{3} + 2\cdot 29^{4} + \left(6 a + 3\right)\cdot 29^{5} + \left(25 a + 24\right)\cdot 29^{6} + \left(6 a + 23\right)\cdot 29^{7} + 4 a\cdot 29^{8} + \left(20 a + 7\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 + 3\cdot 29 + 12\cdot 29^{2} + 21\cdot 29^{3} + 16\cdot 29^{4} + 4\cdot 29^{5} + 28\cdot 29^{6} + 9\cdot 29^{7} + 25\cdot 29^{8} + 8\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)(3,6)$ | $-1$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $6$ | $2$ | $(1,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,5,6)$ | $1$ |
| $6$ | $4$ | $(1,4,5,2)(3,6)$ | $-1$ |
| $8$ | $6$ | $(1,6,4,5,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.