Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 281 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 281 }$: $ x^{2} + 280 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 167 + \left(135 a + 253\right)\cdot 281 + \left(103 a + 256\right)\cdot 281^{2} + \left(260 a + 3\right)\cdot 281^{3} + \left(10 a + 112\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 209 a + 239 + \left(145 a + 35\right)\cdot 281 + \left(177 a + 225\right)\cdot 281^{2} + \left(20 a + 160\right)\cdot 281^{3} + \left(270 a + 143\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 128 + \left(140 a + 132\right)\cdot 281 + \left(128 a + 223\right)\cdot 281^{2} + \left(120 a + 20\right)\cdot 281^{3} + \left(142 a + 238\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 263 a + 146 + \left(140 a + 254\right)\cdot 281 + \left(152 a + 211\right)\cdot 281^{2} + \left(160 a + 12\right)\cdot 281^{3} + \left(138 a + 260\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 65 + \left(135 a + 32\right)\cdot 281 + \left(257 a + 42\right)\cdot 281^{2} + \left(195 a + 72\right)\cdot 281^{3} + \left(202 a + 41\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 248 a + 98 + \left(145 a + 134\right)\cdot 281 + \left(23 a + 164\right)\cdot 281^{2} + \left(85 a + 10\right)\cdot 281^{3} + \left(78 a + 48\right)\cdot 281^{4} +O\left(281^{ 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.
