Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(179776\)\(\medspace = 2^{6} \cdot 53^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.179776.1 |
| 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 6.2.179776.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - x^{2} - x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 5 + 9 a\cdot 31 + \left(25 a + 1\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + \left(30 a + 10\right)\cdot 31^{4} + \left(17 a + 22\right)\cdot 31^{5} + \left(19 a + 18\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 4\cdot 31 + 17\cdot 31^{2} + 15\cdot 31^{3} + 13\cdot 31^{4} + 22\cdot 31^{5} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 2\cdot 31 + 29\cdot 31^{2} + 13\cdot 31^{3} + 20\cdot 31^{4} + 12\cdot 31^{5} + 29\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 14 + \left(21 a + 30\right)\cdot 31 + \left(5 a + 10\right)\cdot 31^{2} + \left(20 a + 22\right)\cdot 31^{3} + 29\cdot 31^{4} + \left(13 a + 27\right)\cdot 31^{5} + \left(11 a + 8\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 19 + \left(21 a + 22\right)\cdot 31 + 11 a\cdot 31^{2} + \left(20 a + 24\right)\cdot 31^{3} + \left(3 a + 15\right)\cdot 31^{4} + \left(3 a + 2\right)\cdot 31^{5} + \left(29 a + 21\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 21 + \left(9 a + 1\right)\cdot 31 + \left(19 a + 3\right)\cdot 31^{2} + \left(10 a + 22\right)\cdot 31^{3} + \left(27 a + 2\right)\cdot 31^{4} + \left(27 a + 5\right)\cdot 31^{5} + \left(a + 14\right)\cdot 31^{6} +O(31^{7})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,6,2)(3,4,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)(5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.