Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(54691\)\(\medspace = 7 \cdot 13 \cdot 601 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.54691.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.54691.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.54691.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{3} + 2x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 35 + \left(33 a + 45\right)\cdot 67 + \left(a + 45\right)\cdot 67^{2} + \left(38 a + 41\right)\cdot 67^{3} + \left(60 a + 24\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a + 20 + \left(12 a + 3\right)\cdot 67 + \left(9 a + 36\right)\cdot 67^{2} + \left(8 a + 28\right)\cdot 67^{3} + \left(39 a + 20\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 a + 17 + \left(33 a + 16\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(28 a + 58\right)\cdot 67^{3} + \left(6 a + 27\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 12 + \left(53 a + 29\right)\cdot 67 + \left(18 a + 9\right)\cdot 67^{2} + \left(23 a + 40\right)\cdot 67^{3} + 57\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 a + 52 + \left(13 a + 30\right)\cdot 67 + \left(48 a + 31\right)\cdot 67^{2} + \left(43 a + 47\right)\cdot 67^{3} + \left(66 a + 35\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 66 + \left(54 a + 8\right)\cdot 67 + \left(57 a + 60\right)\cdot 67^{2} + \left(58 a + 51\right)\cdot 67^{3} + \left(27 a + 34\right)\cdot 67^{4} +O(67^{5})\)
|
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.