Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(294592\)\(\medspace = 2^{6} \cdot 4603 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.294592.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.18412.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.294592.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 4x^{4} - 4x^{3} + x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 97 a + 27 + \left(69 a + 71\right)\cdot 101 + \left(63 a + 79\right)\cdot 101^{2} + \left(18 a + 60\right)\cdot 101^{3} + \left(51 a + 76\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 17 + \left(35 a + 55\right)\cdot 101 + \left(34 a + 58\right)\cdot 101^{2} + \left(66 a + 80\right)\cdot 101^{3} + \left(47 a + 71\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 81 a + 97 + \left(65 a + 74\right)\cdot 101 + \left(66 a + 59\right)\cdot 101^{2} + \left(34 a + 8\right)\cdot 101^{3} + \left(53 a + 95\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 + 86\cdot 101 + 46\cdot 101^{2} + 31\cdot 101^{3} + 90\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 11 + \left(31 a + 52\right)\cdot 101 + \left(37 a + 62\right)\cdot 101^{2} + \left(82 a + 71\right)\cdot 101^{3} + \left(49 a + 60\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 84 + 63\cdot 101 + 96\cdot 101^{2} + 49\cdot 101^{3} + 9\cdot 101^{4} +O(101^{5})\)
|
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.