Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 109 a + 65 + \left(69 a + 123\right)\cdot 127 + \left(80 a + 116\right)\cdot 127^{2} + \left(72 a + 35\right)\cdot 127^{3} + \left(88 a + 89\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 109 a + 103 + \left(20 a + 104\right)\cdot 127 + \left(105 a + 60\right)\cdot 127^{2} + \left(17 a + 25\right)\cdot 127^{3} + \left(110 a + 6\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 47 + \left(57 a + 84\right)\cdot 127 + 46 a\cdot 127^{2} + \left(54 a + 28\right)\cdot 127^{3} + \left(38 a + 105\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 85 + \left(106 a + 16\right)\cdot 127 + \left(21 a + 18\right)\cdot 127^{2} + \left(109 a + 65\right)\cdot 127^{3} + \left(16 a + 98\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 88 + \left(113 a + 49\right)\cdot 127 + \left(124 a + 86\right)\cdot 127^{2} + \left(83 a + 6\right)\cdot 127^{3} + \left(a + 82\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 a + 122 + \left(13 a + 1\right)\cdot 127 + \left(2 a + 98\right)\cdot 127^{2} + \left(43 a + 92\right)\cdot 127^{3} + \left(125 a + 126\right)\cdot 127^{4} +O\left(127^{ 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.
