Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(36235\)\(\medspace = 5 \cdot 7247 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.36235.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.36235.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.36235.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 2 + 16 a\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(14 a + 10\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 19\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 3\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + \left(2 a + 23\right)\cdot 31 + 5 a\cdot 31^{2} + \left(22 a + 12\right)\cdot 31^{3} + \left(29 a + 13\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 10 + \left(28 a + 22\right)\cdot 31 + \left(25 a + 8\right)\cdot 31^{2} + \left(8 a + 20\right)\cdot 31^{3} + \left(a + 19\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 16\cdot 31 + 16\cdot 31^{2} + 2\cdot 31^{3} + 21\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 15 + \left(14 a + 11\right)\cdot 31 + 25 a\cdot 31^{2} + \left(16 a + 2\right)\cdot 31^{3} + \left(24 a + 17\right)\cdot 31^{4} +O(31^{5})\)
|
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.