Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1147\)\(\medspace = 31 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.35557.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.1147.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.42439.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{4} - x^{3} + 2x^{2} - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 45 + \left(14 a + 43\right)\cdot 53 + \left(32 a + 5\right)\cdot 53^{2} + \left(47 a + 37\right)\cdot 53^{3} + 18 a\cdot 53^{4} + \left(33 a + 6\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 42\cdot 53 + 7\cdot 53^{2} + 27\cdot 53^{3} + 26\cdot 53^{4} + 12\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 9\cdot 53 + 39\cdot 53^{2} + 14\cdot 53^{3} + 27\cdot 53^{4} + 26\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 a + 15 + \left(8 a + 5\right)\cdot 53 + \left(9 a + 32\right)\cdot 53^{2} + \left(8 a + 36\right)\cdot 53^{3} + \left(18 a + 5\right)\cdot 53^{4} + \left(44 a + 23\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 11 + \left(44 a + 42\right)\cdot 53 + \left(43 a + 6\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(34 a + 17\right)\cdot 53^{4} + \left(8 a + 23\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 18 + \left(38 a + 16\right)\cdot 53 + \left(20 a + 14\right)\cdot 53^{2} + \left(5 a + 36\right)\cdot 53^{3} + \left(34 a + 28\right)\cdot 53^{4} + \left(19 a + 14\right)\cdot 53^{5} +O(53^{6})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,6,4,5)$ | $1$ |
| $6$ | $4$ | $(1,6,4,5)(2,3)$ | $-1$ |
| $8$ | $6$ | $(1,6,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.