Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(1087849\)\(\medspace = 7^{2} \cdot 149^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.53304601.1 |
| 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 6.2.53304601.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} - 7x^{3} + 19x^{2} + 7x - 15 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 75 + 41\cdot 113 + 108\cdot 113^{3} + 100\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 a + 18 + \left(7 a + 4\right)\cdot 113 + \left(69 a + 93\right)\cdot 113^{2} + \left(a + 62\right)\cdot 113^{3} + \left(39 a + 112\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 107 + 94\cdot 113 + 45\cdot 113^{2} + 26\cdot 113^{3} + 28\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 110 + \left(105 a + 13\right)\cdot 113 + \left(43 a + 10\right)\cdot 113^{2} + \left(111 a + 13\right)\cdot 113^{3} + \left(73 a + 14\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 64 + \left(20 a + 37\right)\cdot 113 + \left(107 a + 83\right)\cdot 113^{2} + \left(64 a + 10\right)\cdot 113^{3} + \left(37 a + 18\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 93 a + 78 + \left(92 a + 33\right)\cdot 113 + \left(5 a + 106\right)\cdot 113^{2} + \left(48 a + 4\right)\cdot 113^{3} + \left(75 a + 65\right)\cdot 113^{4} +O(113^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.