Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(52891\)\(\medspace = 227 \cdot 233 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.52891.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Projective image: | $S_6$ |
Projective field: | Galois closure of 6.0.52891.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ |
\( 16 a + 1 + \left(106 a + 42\right)\cdot 131 + \left(66 a + 50\right)\cdot 131^{2} + \left(104 a + 109\right)\cdot 131^{3} + \left(76 a + 127\right)\cdot 131^{4} +O(131^{5})\)
$r_{ 2 }$ |
$=$ |
\( 108 + 112\cdot 131 + 5\cdot 131^{2} + 120\cdot 131^{3} + 68\cdot 131^{4} +O(131^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 38 a + 87 + \left(76 a + 125\right)\cdot 131 + \left(81 a + 101\right)\cdot 131^{2} + \left(76 a + 12\right)\cdot 131^{3} + \left(119 a + 5\right)\cdot 131^{4} +O(131^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 25 + 55\cdot 131 + 64\cdot 131^{2} + 107\cdot 131^{3} + 108\cdot 131^{4} +O(131^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 93 a + 108 + \left(54 a + 130\right)\cdot 131 + \left(49 a + 89\right)\cdot 131^{2} + \left(54 a + 106\right)\cdot 131^{3} + \left(11 a + 13\right)\cdot 131^{4} +O(131^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 115 a + 65 + \left(24 a + 57\right)\cdot 131 + \left(64 a + 80\right)\cdot 131^{2} + \left(26 a + 67\right)\cdot 131^{3} + \left(54 a + 68\right)\cdot 131^{4} +O(131^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |