Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(190\!\cdots\!121\)\(\medspace = 3^{6} \cdot 7993^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.215811.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T145 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.215811.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} - 2x^{3} + x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$:
\( x^{2} + 192x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 120 a + 144 + \left(128 a + 120\right)\cdot 197 + \left(69 a + 168\right)\cdot 197^{2} + \left(190 a + 107\right)\cdot 197^{3} + \left(32 a + 172\right)\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 142 + 13\cdot 197 + 150\cdot 197^{2} + 8\cdot 197^{3} + 167\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 106 + 75\cdot 197 + 62\cdot 197^{2} + 194\cdot 197^{3} + 194\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a + 52 + \left(196 a + 180\right)\cdot 197 + \left(143 a + 42\right)\cdot 197^{2} + \left(91 a + 177\right)\cdot 197^{3} + \left(45 a + 83\right)\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 169 a + 192 + 147\cdot 197 + \left(53 a + 172\right)\cdot 197^{2} + \left(105 a + 97\right)\cdot 197^{3} + \left(151 a + 22\right)\cdot 197^{4} +O(197^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 a + 153 + \left(68 a + 52\right)\cdot 197 + \left(127 a + 191\right)\cdot 197^{2} + \left(6 a + 4\right)\cdot 197^{3} + \left(164 a + 147\right)\cdot 197^{4} +O(197^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.