Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(856\)\(\medspace = 2^{3} \cdot 107 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.91592.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | odd |
Determinant: | 1.856.2t1.b.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.6848.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 9 a + 36 + \left(46 a + 32\right)\cdot 47 + \left(32 a + 12\right)\cdot 47^{2} + \left(5 a + 4\right)\cdot 47^{3} + \left(16 a + 27\right)\cdot 47^{4} + \left(46 a + 30\right)\cdot 47^{5} +O(47^{6})\) |
$r_{ 2 }$ | $=$ | \( 12 + 37\cdot 47 + 32\cdot 47^{2} + 29\cdot 47^{3} + 46\cdot 47^{4} +O(47^{6})\) |
$r_{ 3 }$ | $=$ | \( 43 + 31\cdot 47 + 33\cdot 47^{2} + 8\cdot 47^{3} + 25\cdot 47^{4} + 16\cdot 47^{5} +O(47^{6})\) |
$r_{ 4 }$ | $=$ | \( 38 a + 7 + 22\cdot 47 + \left(14 a + 32\right)\cdot 47^{2} + \left(41 a + 29\right)\cdot 47^{3} + \left(30 a + 6\right)\cdot 47^{4} + 13\cdot 47^{5} +O(47^{6})\) |
$r_{ 5 }$ | $=$ | \( 14 a + 8 + \left(2 a + 13\right)\cdot 47 + \left(11 a + 28\right)\cdot 47^{2} + \left(32 a + 7\right)\cdot 47^{3} + \left(46 a + 34\right)\cdot 47^{4} + \left(40 a + 45\right)\cdot 47^{5} +O(47^{6})\) |
$r_{ 6 }$ | $=$ | \( 33 a + 36 + \left(44 a + 3\right)\cdot 47 + \left(35 a + 1\right)\cdot 47^{2} + \left(14 a + 14\right)\cdot 47^{3} + 47^{4} + \left(6 a + 34\right)\cdot 47^{5} +O(47^{6})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
$3$ | $2$ | $(1,4)$ | $1$ |
$3$ | $2$ | $(1,4)(5,6)$ | $-1$ |
$6$ | $2$ | $(2,5)(3,6)$ | $1$ |
$6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
$8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
$6$ | $4$ | $(1,6,4,5)$ | $1$ |
$6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
$8$ | $6$ | $(1,6,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.