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:
| $r_{ 1 }$ |
$=$ |
$ 83 a + 106 + \left(30 a + 15\right)\cdot 113 + \left(78 a + 87\right)\cdot 113^{2} + \left(103 a + 111\right)\cdot 113^{3} + \left(97 a + 96\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 85 + \left(82 a + 75\right)\cdot 113 + \left(34 a + 91\right)\cdot 113^{2} + \left(9 a + 34\right)\cdot 113^{3} + \left(15 a + 38\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 74 + 95\cdot 113 + 78\cdot 113^{2} + 72\cdot 113^{3} + 112\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 85 a + 88 + \left(10 a + 98\right)\cdot 113 + \left(11 a + 57\right)\cdot 113^{2} + \left(79 a + 91\right)\cdot 113^{3} + \left(35 a + 46\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 61 + 29\cdot 113 + 87\cdot 113^{2} + 94\cdot 113^{3} + 32\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 106\cdot 113 + 94\cdot 113^{2} + 33\cdot 113^{3} + 67\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 a + 91 + \left(102 a + 29\right)\cdot 113 + \left(101 a + 67\right)\cdot 113^{2} + \left(33 a + 12\right)\cdot 113^{3} + \left(77 a + 57\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$-1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
