Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(60466176\)\(\medspace = 2^{10} \cdot 3^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.120932352.13 |
| 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 6.2.120932352.13 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} - 4x^{3} + 18x^{2} + 12x - 26 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 3\cdot 67 + 26\cdot 67^{2} + 39\cdot 67^{3} + 47\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 44\cdot 67 + 8\cdot 67^{2} + 28\cdot 67^{3} + 20\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 62 + \left(55 a + 41\right)\cdot 67 + \left(48 a + 66\right)\cdot 67^{2} + \left(13 a + 23\right)\cdot 67^{3} + \left(42 a + 36\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 65 a + 3 + \left(11 a + 59\right)\cdot 67 + \left(18 a + 5\right)\cdot 67^{2} + \left(53 a + 30\right)\cdot 67^{3} + \left(24 a + 57\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 53\cdot 67 + 33\cdot 67^{2} + 27\cdot 67^{3} + 10\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 + 66\cdot 67 + 59\cdot 67^{2} + 51\cdot 67^{3} + 28\cdot 67^{4} +O(67^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $5$ | |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ | |
| $15$ | $2$ | $(1,2)$ | $-3$ | |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ | ✓ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | |
| $40$ | $3$ | $(1,2,3)$ | $2$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |