Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(148\!\cdots\!241\)\(\medspace = 33791^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.33791.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.33791.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + x^{3} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + \left(24 a + 44\right)\cdot 47 + \left(6 a + 7\right)\cdot 47^{2} + \left(27 a + 32\right)\cdot 47^{3} + 24\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 a + 10 + \left(22 a + 40\right)\cdot 47 + \left(40 a + 43\right)\cdot 47^{2} + \left(19 a + 32\right)\cdot 47^{3} + \left(46 a + 45\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 6 + \left(24 a + 37\right)\cdot 47 + \left(28 a + 10\right)\cdot 47^{2} + \left(14 a + 32\right)\cdot 47^{3} + \left(7 a + 1\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 + 30\cdot 47 + 11\cdot 47^{2} + 16\cdot 47^{3} + 41\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 9 + \left(22 a + 14\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(32 a + 32\right)\cdot 47^{3} + \left(39 a + 1\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 21\cdot 47 + 23\cdot 47^{2} + 41\cdot 47^{3} + 25\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.