Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(428\!\cdots\!241\)\(\medspace = 109^{4} \cdot 2347^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.255823.1 |
| 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 6.4.255823.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} + 3x^{3} - 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 82 + \left(63 a + 43\right)\cdot 103 + \left(33 a + 10\right)\cdot 103^{2} + \left(73 a + 44\right)\cdot 103^{3} + \left(18 a + 84\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 82 + \left(102 a + 88\right)\cdot 103 + \left(24 a + 99\right)\cdot 103^{2} + \left(40 a + 67\right)\cdot 103^{3} + \left(99 a + 90\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 96 a + 89 + 80\cdot 103 + \left(78 a + 22\right)\cdot 103^{2} + \left(62 a + 83\right)\cdot 103^{3} + \left(3 a + 46\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 70 a + 12 + \left(39 a + 74\right)\cdot 103 + \left(69 a + 83\right)\cdot 103^{2} + \left(29 a + 83\right)\cdot 103^{3} + \left(84 a + 29\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 + 54\cdot 103 + 15\cdot 103^{2} + 80\cdot 103^{3} + 93\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 70\cdot 103 + 76\cdot 103^{2} + 52\cdot 103^{3} + 66\cdot 103^{4} +O(103^{5})\)
|
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.