Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(938809600\)\(\medspace = 2^{8} \cdot 5^{2} \cdot 383^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.122560.1 |
| 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 6.2.122560.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 4x^{4} - 4x^{3} + 3x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 4 + \left(13 a + 21\right)\cdot 73 + \left(53 a + 28\right)\cdot 73^{2} + \left(30 a + 24\right)\cdot 73^{3} + \left(19 a + 14\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 a + 70 + \left(56 a + 13\right)\cdot 73 + \left(12 a + 57\right)\cdot 73^{2} + \left(50 a + 35\right)\cdot 73^{3} + \left(40 a + 36\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a + 17 + \left(16 a + 6\right)\cdot 73 + \left(60 a + 39\right)\cdot 73^{2} + \left(22 a + 27\right)\cdot 73^{3} + \left(32 a + 35\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 a + 25 + \left(59 a + 53\right)\cdot 73 + \left(19 a + 28\right)\cdot 73^{2} + \left(42 a + 63\right)\cdot 73^{3} + \left(53 a + 41\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 51 + \left(59 a + 10\right)\cdot 73 + \left(41 a + 36\right)\cdot 73^{2} + \left(27 a + 13\right)\cdot 73^{3} + \left(51 a + 55\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 72 a + 54 + \left(13 a + 40\right)\cdot 73 + \left(31 a + 29\right)\cdot 73^{2} + \left(45 a + 54\right)\cdot 73^{3} + \left(21 a + 35\right)\cdot 73^{4} +O(73^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.