Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(41489609581920256\)\(\medspace = 2^{24} \cdot 223^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.228352.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.228352.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 4x^{3} - 4x^{2} + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 80 + 69\cdot 191 + 27\cdot 191^{2} + 131\cdot 191^{3} + 20\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 a + 39 + \left(113 a + 96\right)\cdot 191 + \left(93 a + 126\right)\cdot 191^{2} + \left(108 a + 17\right)\cdot 191^{3} + \left(80 a + 158\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 174\cdot 191 + 131\cdot 191^{2} + 2\cdot 191^{3} + 176\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 54 + \left(71 a + 108\right)\cdot 191 + \left(65 a + 188\right)\cdot 191^{2} + \left(5 a + 128\right)\cdot 191^{3} + \left(59 a + 112\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 180 a + 65 + \left(119 a + 168\right)\cdot 191 + \left(125 a + 182\right)\cdot 191^{2} + \left(185 a + 68\right)\cdot 191^{3} + \left(131 a + 166\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 129 a + 101 + \left(77 a + 147\right)\cdot 191 + \left(97 a + 106\right)\cdot 191^{2} + \left(82 a + 32\right)\cdot 191^{3} + \left(110 a + 130\right)\cdot 191^{4} +O(191^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.