Basic invariants
Galois action
Roots of defining polynomial
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} + 269 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 110 + 20\cdot 271 + 12\cdot 271^{2} + 45\cdot 271^{3} + 18\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 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\left(271^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 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\left(271^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 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\left(271^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 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\left(271^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 252 + 6\cdot 271 + 229\cdot 271^{2} + 50\cdot 271^{3} + 217\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $15$ |
$2$ |
$(1,2)$ |
$-3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
