Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(50587\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.50587.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.50587.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.50587.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 2x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 271 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 271 }$:
\( x^{2} + 269x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 110 + 20\cdot 271 + 12\cdot 271^{2} + 45\cdot 271^{3} + 18\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 242 a + 72 + \left(142 a + 93\right)\cdot 271 + \left(162 a + 47\right)\cdot 271^{2} + \left(47 a + 183\right)\cdot 271^{3} + \left(130 a + 142\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 183 a + \left(199 a + 34\right)\cdot 271 + \left(33 a + 78\right)\cdot 271^{2} + \left(7 a + 83\right)\cdot 271^{3} + \left(5 a + 38\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 88 a + 95 + \left(71 a + 250\right)\cdot 271 + \left(237 a + 216\right)\cdot 271^{2} + \left(263 a + 63\right)\cdot 271^{3} + \left(265 a + 41\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 a + 14 + \left(128 a + 137\right)\cdot 271 + \left(108 a + 229\right)\cdot 271^{2} + \left(223 a + 115\right)\cdot 271^{3} + \left(140 a + 84\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 252 + 6\cdot 271 + 229\cdot 271^{2} + 50\cdot 271^{3} + 217\cdot 271^{4} +O(271^{5})\)
|
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.