Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(811\!\cdots\!016\)\(\medspace = 2^{24} \cdot 3851^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.30808.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1252 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.30808.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{4} - x^{3} + x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 a + 99 + \left(77 a + 33\right)\cdot 109 + 108 a\cdot 109^{2} + \left(71 a + 26\right)\cdot 109^{3} + \left(84 a + 78\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 102 + \left(29 a + 107\right)\cdot 109 + \left(67 a + 17\right)\cdot 109^{2} + \left(18 a + 7\right)\cdot 109^{3} + \left(9 a + 77\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 89 a + 13 + \left(79 a + 8\right)\cdot 109 + \left(41 a + 56\right)\cdot 109^{2} + \left(90 a + 67\right)\cdot 109^{3} + \left(99 a + 67\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 a + 49 + \left(31 a + 52\right)\cdot 109 + 31\cdot 109^{2} + \left(37 a + 98\right)\cdot 109^{3} + \left(24 a + 90\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 85 + \left(62 a + 32\right)\cdot 109 + \left(80 a + 101\right)\cdot 109^{2} + \left(3 a + 47\right)\cdot 109^{3} + \left(83 a + 21\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 106 a + 88 + \left(46 a + 91\right)\cdot 109 + \left(28 a + 10\right)\cdot 109^{2} + \left(105 a + 80\right)\cdot 109^{3} + \left(25 a + 100\right)\cdot 109^{4} +O(109^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.