Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 283 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 283 }$: $ x^{2} + 282 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 162 a + 127 + \left(122 a + 122\right)\cdot 283 + \left(251 a + 272\right)\cdot 283^{2} + \left(191 a + 65\right)\cdot 283^{3} + \left(10 a + 270\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 229 + \left(235 a + 92\right)\cdot 283 + \left(128 a + 131\right)\cdot 283^{2} + \left(41 a + 41\right)\cdot 283^{3} + \left(137 a + 152\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 268 a + 124 + \left(269 a + 123\right)\cdot 283 + \left(149 a + 69\right)\cdot 283^{2} + \left(35 a + 23\right)\cdot 283^{3} + \left(173 a + 259\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 109 + \left(13 a + 125\right)\cdot 283 + \left(133 a + 232\right)\cdot 283^{2} + \left(247 a + 191\right)\cdot 283^{3} + \left(109 a + 113\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 121 a + 6 + \left(160 a + 83\right)\cdot 283 + \left(31 a + 118\right)\cdot 283^{2} + \left(91 a + 6\right)\cdot 283^{3} + \left(272 a + 89\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 257 a + 255 + \left(47 a + 18\right)\cdot 283 + \left(154 a + 25\right)\cdot 283^{2} + \left(241 a + 237\right)\cdot 283^{3} + \left(145 a + 247\right)\cdot 283^{4} +O\left(283^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
