Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(6241849278890625\)\(\medspace = 3^{14} \cdot 5^{6} \cdot 17^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.309825.1 |
| 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 6.2.309825.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{3} - 3x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 + 21\cdot 43 + 41\cdot 43^{2} + 26\cdot 43^{3} + 10\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 8 + \left(36 a + 4\right)\cdot 43 + \left(3 a + 19\right)\cdot 43^{2} + \left(23 a + 7\right)\cdot 43^{3} + \left(23 a + 15\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 36 + \left(30 a + 9\right)\cdot 43 + \left(13 a + 9\right)\cdot 43^{2} + \left(9 a + 24\right)\cdot 43^{3} + \left(8 a + 6\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 + 28\cdot 43 + 36\cdot 43^{2} + 23\cdot 43^{3} + 32\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 a + 15 + \left(6 a + 33\right)\cdot 43 + \left(39 a + 29\right)\cdot 43^{2} + \left(19 a + 26\right)\cdot 43^{3} + \left(19 a + 15\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a + 2 + \left(12 a + 31\right)\cdot 43 + \left(29 a + 35\right)\cdot 43^{2} + \left(33 a + 19\right)\cdot 43^{3} + \left(34 a + 5\right)\cdot 43^{4} +O(43^{5})\)
|
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.