Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(1680066179072\)\(\medspace = 2^{12} \cdot 743^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.47552.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | odd |
Determinant: | 1.743.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.47552.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{4} - 2x^{3} + x^{2} + 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 128 + 104\cdot 131 + 67\cdot 131^{2} + 47\cdot 131^{3} + 2\cdot 131^{4} +O(131^{5})\)
$r_{ 2 }$ |
$=$ |
\( 18 a + 12 + \left(101 a + 56\right)\cdot 131 + \left(67 a + 25\right)\cdot 131^{2} + \left(98 a + 127\right)\cdot 131^{3} + \left(63 a + 119\right)\cdot 131^{4} +O(131^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 113 a + 84 + \left(29 a + 49\right)\cdot 131 + \left(63 a + 64\right)\cdot 131^{2} + \left(32 a + 60\right)\cdot 131^{3} + \left(67 a + 14\right)\cdot 131^{4} +O(131^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a + 81 + \left(57 a + 10\right)\cdot 131 + \left(6 a + 104\right)\cdot 131^{2} + \left(64 a + 97\right)\cdot 131^{3} + \left(12 a + 51\right)\cdot 131^{4} +O(131^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 116 + 69\cdot 131 + 58\cdot 131^{2} + 105\cdot 131^{3} + 35\cdot 131^{4} +O(131^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 125 a + 105 + \left(73 a + 101\right)\cdot 131 + \left(124 a + 72\right)\cdot 131^{2} + \left(66 a + 85\right)\cdot 131^{3} + \left(118 a + 37\right)\cdot 131^{4} +O(131^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $-1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.