Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(3013696\)\(\medspace = 2^{6} \cdot 7^{2} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.3013696.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.2.3013696.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 2x^{3} - x^{2} + 6x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$:
\( x^{2} + 131x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 133\cdot 137 + 23\cdot 137^{2} + 120\cdot 137^{3} + 75\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 88 a + 19 + \left(119 a + 49\right)\cdot 137 + \left(95 a + 82\right)\cdot 137^{2} + \left(93 a + 123\right)\cdot 137^{3} + \left(82 a + 36\right)\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 a + 136 + \left(17 a + 130\right)\cdot 137 + \left(41 a + 126\right)\cdot 137^{2} + \left(43 a + 41\right)\cdot 137^{3} + \left(54 a + 28\right)\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 126 a + 6 + \left(61 a + 124\right)\cdot 137 + \left(2 a + 59\right)\cdot 137^{2} + \left(69 a + 105\right)\cdot 137^{3} + \left(105 a + 7\right)\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 77 + \left(75 a + 95\right)\cdot 137 + \left(134 a + 12\right)\cdot 137^{2} + \left(67 a + 106\right)\cdot 137^{3} + \left(31 a + 23\right)\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 + 15\cdot 137 + 105\cdot 137^{2} + 50\cdot 137^{3} + 101\cdot 137^{4} +O(137^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.