Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(510\!\cdots\!681\)\(\medspace = 29077^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.29077.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1252 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.29077.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - x^{2} + 2x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 8 a + 129 + \left(55 a + 128\right)\cdot 131 + \left(78 a + 32\right)\cdot 131^{2} + \left(19 a + 108\right)\cdot 131^{3} + \left(57 a + 50\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 123 a + 30 + \left(75 a + 79\right)\cdot 131 + \left(52 a + 29\right)\cdot 131^{2} + \left(111 a + 108\right)\cdot 131^{3} + \left(73 a + 128\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 111 a + 49 + \left(54 a + 105\right)\cdot 131 + \left(61 a + 111\right)\cdot 131^{2} + \left(18 a + 53\right)\cdot 131^{3} + \left(13 a + 4\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 55 a + 64 + \left(70 a + 81\right)\cdot 131 + \left(109 a + 101\right)\cdot 131^{2} + \left(126 a + 25\right)\cdot 131^{3} + \left(24 a + 33\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 a + 22 + \left(60 a + 46\right)\cdot 131 + \left(21 a + 76\right)\cdot 131^{2} + \left(4 a + 30\right)\cdot 131^{3} + \left(106 a + 6\right)\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 100 + \left(76 a + 82\right)\cdot 131 + \left(69 a + 40\right)\cdot 131^{2} + \left(112 a + 66\right)\cdot 131^{3} + \left(117 a + 38\right)\cdot 131^{4} +O(131^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.