Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 129 + \left(55 a + 128\right)\cdot 131 + \left(78 a + 32\right)\cdot 131^{2} + \left(19 a + 108\right)\cdot 131^{3} + \left(57 a + 50\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 123 a + 30 + \left(75 a + 79\right)\cdot 131 + \left(52 a + 29\right)\cdot 131^{2} + \left(111 a + 108\right)\cdot 131^{3} + \left(73 a + 128\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 111 a + 49 + \left(54 a + 105\right)\cdot 131 + \left(61 a + 111\right)\cdot 131^{2} + \left(18 a + 53\right)\cdot 131^{3} + \left(13 a + 4\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a + 64 + \left(70 a + 81\right)\cdot 131 + \left(109 a + 101\right)\cdot 131^{2} + \left(126 a + 25\right)\cdot 131^{3} + \left(24 a + 33\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 a + 22 + \left(60 a + 46\right)\cdot 131 + \left(21 a + 76\right)\cdot 131^{2} + \left(4 a + 30\right)\cdot 131^{3} + \left(106 a + 6\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 100 + \left(76 a + 82\right)\cdot 131 + \left(69 a + 40\right)\cdot 131^{2} + \left(112 a + 66\right)\cdot 131^{3} + \left(117 a + 38\right)\cdot 131^{4} +O\left(131^{ 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.
