Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(45203\)\(\medspace = 17 \cdot 2659 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.45203.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.45203.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.45203.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 34 + \left(30 a + 27\right)\cdot 53 + \left(13 a + 48\right)\cdot 53^{2} + \left(23 a + 30\right)\cdot 53^{3} + \left(47 a + 3\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 44 + 36\cdot 53 + 39\cdot 53^{2} + 11\cdot 53^{3} + 13\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a + 45 + \left(22 a + 26\right)\cdot 53 + \left(39 a + 19\right)\cdot 53^{2} + \left(29 a + 4\right)\cdot 53^{3} + \left(5 a + 11\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 5\cdot 53 + 28\cdot 53^{2} + 26\cdot 53^{3} + 10\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 a + 45 + \left(44 a + 43\right)\cdot 53 + \left(31 a + 49\right)\cdot 53^{2} + \left(48 a + 40\right)\cdot 53^{3} + \left(40 a + 2\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 17 + \left(8 a + 18\right)\cdot 53 + \left(21 a + 26\right)\cdot 53^{2} + \left(4 a + 44\right)\cdot 53^{3} + \left(12 a + 11\right)\cdot 53^{4} +O(53^{5})\)
|
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.