Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(6141217316608457521\)\(\medspace = 67^{4} \cdot 743^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.49781.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T183 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.49781.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 25 + \left(27 a + 2\right)\cdot 41 + \left(a + 36\right)\cdot 41^{2} + \left(15 a + 32\right)\cdot 41^{3} + \left(5 a + 32\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 36 + \left(22 a + 26\right)\cdot 41 + \left(34 a + 1\right)\cdot 41^{2} + \left(39 a + 6\right)\cdot 41^{3} + \left(30 a + 5\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 35 + \left(13 a + 26\right)\cdot 41 + \left(39 a + 13\right)\cdot 41^{2} + \left(25 a + 35\right)\cdot 41^{3} + \left(35 a + 33\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 28 + 18 a\cdot 41 + \left(6 a + 1\right)\cdot 41^{2} + \left(a + 9\right)\cdot 41^{3} + \left(10 a + 17\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 25 + \left(24 a + 28\right)\cdot 41 + \left(38 a + 30\right)\cdot 41^{2} + \left(19 a + 29\right)\cdot 41^{3} + \left(34 a + 36\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 15 + \left(16 a + 37\right)\cdot 41 + \left(2 a + 39\right)\cdot 41^{2} + \left(21 a + 9\right)\cdot 41^{3} + \left(6 a + 38\right)\cdot 41^{4} +O(41^{5})\)
|
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.