Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 69 a + 96 + \left(38 a + 104\right)\cdot 107 + \left(41 a + 19\right)\cdot 107^{2} + \left(93 a + 12\right)\cdot 107^{3} + \left(26 a + 96\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 a + 51 + \left(68 a + 83\right)\cdot 107 + \left(65 a + 39\right)\cdot 107^{2} + \left(13 a + 23\right)\cdot 107^{3} + \left(80 a + 3\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 105 + 43\cdot 107 + 42\cdot 107^{2} + 84\cdot 107^{3} + 59\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 50\cdot 107 + 40\cdot 107^{2} + 35\cdot 107^{3} + 7\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 43\cdot 107 + 89\cdot 107^{2} + 60\cdot 107^{3} + 39\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 a + 78 + \left(78 a + 86\right)\cdot 107 + \left(a + 26\right)\cdot 107^{2} + \left(89 a + 89\right)\cdot 107^{3} + \left(10 a + 26\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 a + 17 + \left(28 a + 15\right)\cdot 107 + \left(105 a + 62\right)\cdot 107^{2} + \left(17 a + 15\right)\cdot 107^{3} + \left(96 a + 88\right)\cdot 107^{4} +O\left(107^{ 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.
