Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(302\!\cdots\!361\)\(\medspace = 29^{4} \cdot 4547^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.131863.1 |
| 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 6.0.131863.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} - 3x^{3} + 4x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 80 a + 36 + \left(104 a + 86\right)\cdot 113 + \left(35 a + 76\right)\cdot 113^{2} + \left(13 a + 56\right)\cdot 113^{3} + \left(83 a + 31\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 59 a + 15 + \left(103 a + 95\right)\cdot 113 + \left(16 a + 79\right)\cdot 113^{2} + \left(91 a + 87\right)\cdot 113^{3} + \left(30 a + 83\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 92 + \left(8 a + 19\right)\cdot 113 + \left(77 a + 64\right)\cdot 113^{2} + \left(99 a + 67\right)\cdot 113^{3} + \left(29 a + 111\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 a + 45 + \left(9 a + 35\right)\cdot 113 + \left(96 a + 66\right)\cdot 113^{2} + \left(21 a + 34\right)\cdot 113^{3} + \left(82 a + 23\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 100\cdot 113 + 110\cdot 113^{2} + 52\cdot 113^{3} + 84\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 + 2\cdot 113 + 54\cdot 113^{2} + 39\cdot 113^{3} + 4\cdot 113^{4} +O(113^{5})\)
|
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.