Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(8415099419290201\)\(\medspace = 11^{6} \cdot 41^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.2237411.1 |
| 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 6.4.2237411.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 4x^{4} + 6x^{3} - 6x + 5 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 30 a + 38 + 14\cdot 53 + \left(48 a + 3\right)\cdot 53^{2} + \left(40 a + 43\right)\cdot 53^{3} + 4 a\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 52 + \left(52 a + 39\right)\cdot 53 + \left(4 a + 35\right)\cdot 53^{2} + \left(12 a + 52\right)\cdot 53^{3} + \left(48 a + 31\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 52\cdot 53 + 30\cdot 53^{2} + 48\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 a + 40 + \left(12 a + 26\right)\cdot 53 + \left(33 a + 39\right)\cdot 53^{2} + \left(6 a + 40\right)\cdot 53^{3} + \left(52 a + 52\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 32 + \left(40 a + 27\right)\cdot 53 + 19 a\cdot 53^{2} + \left(46 a + 34\right)\cdot 53^{3} + 42\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 50\cdot 53 + 48\cdot 53^{2} + 45\cdot 53^{3} + 32\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.