Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(36107\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.36107.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.36107.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.36107.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$:
\( x^{2} + 127x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 57 a + 61 + \left(110 a + 39\right)\cdot 131 + \left(68 a + 97\right)\cdot 131^{2} + \left(2 a + 75\right)\cdot 131^{3} + \left(80 a + 56\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 74 a + 27 + \left(20 a + 31\right)\cdot 131 + 62 a\cdot 131^{2} + \left(128 a + 17\right)\cdot 131^{3} + \left(50 a + 112\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 62 a + 109 + \left(56 a + 108\right)\cdot 131 + \left(97 a + 94\right)\cdot 131^{2} + \left(29 a + 76\right)\cdot 131^{3} + \left(53 a + 99\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 a + 95 + \left(74 a + 10\right)\cdot 131 + \left(33 a + 35\right)\cdot 131^{2} + \left(101 a + 98\right)\cdot 131^{3} + \left(77 a + 20\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 + 92\cdot 131 + 97\cdot 131^{2} + 96\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 79 + 110\cdot 131 + 67\cdot 131^{2} + 28\cdot 131^{3} + 12\cdot 131^{4} +O(131^{5})\)
|
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.