Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(159028\)\(\medspace = 2^{2} \cdot 83 \cdot 479 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.159028.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.159028.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.159028.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 a + 42 + \left(11 a + 48\right)\cdot 73 + \left(30 a + 24\right)\cdot 73^{2} + \left(59 a + 27\right)\cdot 73^{3} + \left(45 a + 46\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 a + 62 + \left(61 a + 51\right)\cdot 73 + \left(42 a + 30\right)\cdot 73^{2} + \left(13 a + 29\right)\cdot 73^{3} + \left(27 a + 51\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 2 + \left(23 a + 46\right)\cdot 73 + \left(35 a + 25\right)\cdot 73^{2} + \left(11 a + 69\right)\cdot 73^{3} + \left(8 a + 38\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 16 + 2 a\cdot 73 + \left(40 a + 29\right)\cdot 73^{2} + \left(23 a + 33\right)\cdot 73^{3} + \left(54 a + 18\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 72 + \left(70 a + 37\right)\cdot 73 + 32 a\cdot 73^{2} + \left(49 a + 64\right)\cdot 73^{3} + \left(18 a + 11\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 65 a + 26 + \left(49 a + 34\right)\cdot 73 + \left(37 a + 35\right)\cdot 73^{2} + \left(61 a + 68\right)\cdot 73^{3} + \left(64 a + 51\right)\cdot 73^{4} +O(73^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.