Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(75559\)\(\medspace = 11 \cdot 6869 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.75559.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.75559.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.75559.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} - 2x^{3} + 4x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 44 + \left(6 a + 3\right)\cdot 59 + \left(39 a + 21\right)\cdot 59^{2} + \left(7 a + 17\right)\cdot 59^{3} + \left(29 a + 10\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 42\cdot 59 + 33\cdot 59^{2} + 30\cdot 59^{3} + 55\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 a + 13 + \left(3 a + 55\right)\cdot 59 + \left(4 a + 24\right)\cdot 59^{2} + \left(25 a + 43\right)\cdot 59^{3} + \left(51 a + 49\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 49 + \left(55 a + 22\right)\cdot 59 + \left(54 a + 25\right)\cdot 59^{2} + \left(33 a + 5\right)\cdot 59^{3} + \left(7 a + 17\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 58\cdot 59 + 17\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 a + 1 + \left(52 a + 53\right)\cdot 59 + \left(19 a + 53\right)\cdot 59^{2} + \left(51 a + 44\right)\cdot 59^{3} + \left(29 a + 31\right)\cdot 59^{4} +O(59^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.