Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(44423\)\(\medspace = 31 \cdot 1433 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.44423.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.44423.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.44423.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{4} - x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 103 a + 58 + \left(50 a + 127\right)\cdot 131 + \left(115 a + 80\right)\cdot 131^{2} + \left(43 a + 79\right)\cdot 131^{3} + \left(90 a + 68\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 2 }$ | $=$ | \( 80 + 67\cdot 131 + 124\cdot 131^{2} + 83\cdot 131^{3} + 98\cdot 131^{4} +O(131^{5})\) |
$r_{ 3 }$ | $=$ | \( 49 a + 120 + \left(41 a + 28\right)\cdot 131 + \left(99 a + 41\right)\cdot 131^{2} + \left(41 a + 38\right)\cdot 131^{3} + \left(127 a + 8\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 + 58\cdot 131 + 43\cdot 131^{2} + 76\cdot 131^{3} + 10\cdot 131^{4} +O(131^{5})\) |
$r_{ 5 }$ | $=$ | \( 82 a + 54 + \left(89 a + 14\right)\cdot 131 + \left(31 a + 4\right)\cdot 131^{2} + \left(89 a + 106\right)\cdot 131^{3} + \left(3 a + 82\right)\cdot 131^{4} +O(131^{5})\) |
$r_{ 6 }$ | $=$ | \( 28 a + 77 + \left(80 a + 96\right)\cdot 131 + \left(15 a + 98\right)\cdot 131^{2} + \left(87 a + 8\right)\cdot 131^{3} + \left(40 a + 124\right)\cdot 131^{4} +O(131^{5})\) |
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.