Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(258\!\cdots\!281\)\(\medspace = 17^{4} \cdot 13259^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.225403.1 |
| 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 6.4.225403.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 4x^{4} + 4x^{3} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 72 + \left(68 a + 74\right)\cdot 83 + \left(60 a + 70\right)\cdot 83^{2} + \left(45 a + 70\right)\cdot 83^{3} + \left(73 a + 30\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 77 a + 78 + \left(14 a + 53\right)\cdot 83 + \left(22 a + 63\right)\cdot 83^{2} + \left(37 a + 55\right)\cdot 83^{3} + \left(9 a + 58\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 44\cdot 83 + 11\cdot 83^{2} + 3\cdot 83^{3} + 45\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 52 + \left(42 a + 41\right)\cdot 83 + \left(74 a + 7\right)\cdot 83^{2} + \left(46 a + 33\right)\cdot 83^{3} + \left(35 a + 33\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 76\cdot 83 + 55\cdot 83^{2} + 80\cdot 83^{3} + 58\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a + 12 + \left(40 a + 41\right)\cdot 83 + \left(8 a + 39\right)\cdot 83^{2} + \left(36 a + 5\right)\cdot 83^{3} + \left(47 a + 22\right)\cdot 83^{4} +O(83^{5})\)
|
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.