Basic invariants
| Dimension: | $10$ |
| Group: | $A_6$ |
| Conductor: | \(20100618201669201\)\(\medspace = 3^{20} \cdot 7^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.15752961.1 |
| 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 6.2.15752961.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 3x^{4} + 9x^{3} - 18x^{2} - 9x + 18 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{2} + 149x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 43 + \left(58 a + 148\right)\cdot 151 + \left(49 a + 52\right)\cdot 151^{2} + \left(129 a + 106\right)\cdot 151^{3} + \left(108 a + 148\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 a + 76 + \left(8 a + 37\right)\cdot 151 + \left(104 a + 23\right)\cdot 151^{2} + \left(108 a + 76\right)\cdot 151^{3} + \left(150 a + 133\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 148 a + 49 + \left(92 a + 110\right)\cdot 151 + \left(101 a + 93\right)\cdot 151^{2} + \left(21 a + 13\right)\cdot 151^{3} + \left(42 a + 86\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 123 + 50\cdot 151 + 86\cdot 151^{2} + 128\cdot 151^{3} + 24\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 103 a + 21 + \left(142 a + 6\right)\cdot 151 + \left(46 a + 72\right)\cdot 151^{2} + \left(42 a + 38\right)\cdot 151^{3} + 24\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 144 + 99\cdot 151 + 124\cdot 151^{2} + 89\cdot 151^{3} + 35\cdot 151^{4} +O(151^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.