Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(3196661779891\)\(\medspace = 14731^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.14731.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | odd |
Determinant: | 1.14731.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.14731.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{3} - x^{2} + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 21 + 42\cdot 47 + 46\cdot 47^{2} + 10\cdot 47^{3} + 41\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 + 18\cdot 47 + 28\cdot 47^{2} + 9\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 15 a + 10 + \left(29 a + 17\right)\cdot 47 + \left(46 a + 29\right)\cdot 47^{2} + \left(12 a + 40\right)\cdot 47^{3} + \left(38 a + 3\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 42 a + 11 + \left(7 a + 14\right)\cdot 47 + \left(8 a + 14\right)\cdot 47^{2} + \left(45 a + 12\right)\cdot 47^{3} + \left(26 a + 30\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 32 a + 40 + \left(17 a + 13\right)\cdot 47 + 46\cdot 47^{2} + \left(34 a + 19\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 5 a + 1 + \left(39 a + 35\right)\cdot 47 + \left(38 a + 22\right)\cdot 47^{2} + a\cdot 47^{3} + \left(20 a + 39\right)\cdot 47^{4} +O(47^{5})\) |
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 |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $-1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.