Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 92 a + 87 + \left(91 a + 82\right)\cdot 113 + \left(21 a + 110\right)\cdot 113^{2} + \left(101 a + 14\right)\cdot 113^{3} + \left(26 a + 34\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 61 + \left(21 a + 75\right)\cdot 113 + \left(91 a + 54\right)\cdot 113^{2} + \left(11 a + 77\right)\cdot 113^{3} + \left(86 a + 29\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 76 + 13\cdot 113 + 46\cdot 113^{2} + 81\cdot 113^{3} + 68\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 12 + \left(5 a + 33\right)\cdot 113 + \left(22 a + 41\right)\cdot 113^{2} + \left(3 a + 16\right)\cdot 113^{3} + \left(9 a + 52\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 + 90\cdot 113 + 77\cdot 113^{2} + 12\cdot 113^{3} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 + 84\cdot 113 + 46\cdot 113^{2} + 103\cdot 113^{3} + 109\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 89 a + 74 + \left(107 a + 71\right)\cdot 113 + \left(90 a + 74\right)\cdot 113^{2} + \left(109 a + 32\right)\cdot 113^{3} + \left(103 a + 44\right)\cdot 113^{4} +O\left(113^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$35$ |
| $21$ |
$2$ |
$(1,2)$ |
$-5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
