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:
| $r_{ 1 }$ |
$=$ |
$ 84 + 126\cdot 127 + 45\cdot 127^{2} + 67\cdot 127^{3} + 90\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 38 + \left(125 a + 79\right)\cdot 127 + 28 a\cdot 127^{2} + \left(68 a + 53\right)\cdot 127^{3} + \left(55 a + 84\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 60 + \left(88 a + 93\right)\cdot 127 + \left(62 a + 73\right)\cdot 127^{2} + \left(61 a + 57\right)\cdot 127^{3} + \left(28 a + 97\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 120 a + 45 + \left(a + 70\right)\cdot 127 + \left(98 a + 31\right)\cdot 127^{2} + \left(58 a + 92\right)\cdot 127^{3} + \left(71 a + 71\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 a + 113 + \left(38 a + 1\right)\cdot 127 + \left(64 a + 48\right)\cdot 127^{2} + \left(65 a + 56\right)\cdot 127^{3} + \left(98 a + 64\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 + 62\cdot 127 + 85\cdot 127^{2} + 24\cdot 127^{3} + 75\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 + 73\cdot 127 + 95\cdot 127^{2} + 29\cdot 127^{3} + 24\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
