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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 112 + \left(96 a + 78\right)\cdot 113 + \left(106 a + 58\right)\cdot 113^{2} + \left(100 a + 65\right)\cdot 113^{3} + \left(8 a + 52\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 44\cdot 113 + 63\cdot 113^{2} + 61\cdot 113^{3} + 25\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 a + 1 + \left(20 a + 105\right)\cdot 113 + \left(9 a + 15\right)\cdot 113^{2} + \left(16 a + 94\right)\cdot 113^{3} + \left(5 a + 13\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 25 + \left(83 a + 65\right)\cdot 113 + \left(11 a + 74\right)\cdot 113^{2} + \left(84 a + 22\right)\cdot 113^{3} + \left(4 a + 78\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 a + 3 + \left(16 a + 67\right)\cdot 113 + \left(6 a + 1\right)\cdot 113^{2} + \left(12 a + 40\right)\cdot 113^{3} + \left(104 a + 58\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 84 a + 34 + \left(29 a + 18\right)\cdot 113 + \left(101 a + 19\right)\cdot 113^{2} + \left(28 a + 3\right)\cdot 113^{3} + \left(108 a + 51\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 a + 48 + \left(92 a + 73\right)\cdot 113 + \left(103 a + 105\right)\cdot 113^{2} + \left(96 a + 51\right)\cdot 113^{3} + \left(107 a + 59\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 value |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
