Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(630\!\cdots\!561\)\(\medspace = 17^{12} \cdot 101^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.29189.1 |
| 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 6.2.29189.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + x^{3} - 2x^{2} + x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 97 a + 59 + \left(40 a + 17\right)\cdot 109 + \left(86 a + 40\right)\cdot 109^{2} + \left(2 a + 24\right)\cdot 109^{3} + \left(15 a + 40\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 80 + \left(a + 76\right)\cdot 109 + \left(92 a + 36\right)\cdot 109^{2} + \left(3 a + 18\right)\cdot 109^{3} + \left(82 a + 75\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 47 + \left(68 a + 70\right)\cdot 109 + \left(22 a + 85\right)\cdot 109^{2} + \left(106 a + 49\right)\cdot 109^{3} + \left(93 a + 52\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 93 a + 37 + \left(33 a + 74\right)\cdot 109 + \left(3 a + 33\right)\cdot 109^{2} + \left(82 a + 58\right)\cdot 109^{3} + \left(51 a + 72\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 104 a + 85 + \left(107 a + 72\right)\cdot 109 + \left(16 a + 18\right)\cdot 109^{2} + \left(105 a + 39\right)\cdot 109^{3} + \left(26 a + 44\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 21 + \left(75 a + 15\right)\cdot 109 + \left(105 a + 3\right)\cdot 109^{2} + \left(26 a + 28\right)\cdot 109^{3} + \left(57 a + 42\right)\cdot 109^{4} +O(109^{5})\)
|
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.