Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 55 + \left(80 a + 50\right)\cdot 83 + \left(8 a + 1\right)\cdot 83^{2} + \left(21 a + 53\right)\cdot 83^{3} + \left(17 a + 13\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 75 a + 22 + \left(17 a + 34\right)\cdot 83 + \left(81 a + 32\right)\cdot 83^{2} + \left(15 a + 43\right)\cdot 83^{3} + \left(55 a + 52\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 58 + \left(15 a + 45\right)\cdot 83 + \left(66 a + 53\right)\cdot 83^{2} + \left(33 a + 76\right)\cdot 83^{3} + \left(5 a + 33\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 67\cdot 83 + 30\cdot 83^{2} + 71\cdot 83^{3} + 41\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 a + 81 + \left(67 a + 37\right)\cdot 83 + \left(16 a + 21\right)\cdot 83^{2} + \left(49 a + 44\right)\cdot 83^{3} + \left(77 a + 5\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a + 67 + \left(2 a + 35\right)\cdot 83 + \left(74 a + 13\right)\cdot 83^{2} + \left(61 a + 65\right)\cdot 83^{3} + \left(65 a + 9\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 8 a + 14 + \left(65 a + 60\right)\cdot 83 + \left(a + 12\right)\cdot 83^{2} + \left(67 a + 61\right)\cdot 83^{3} + \left(27 a + 8\right)\cdot 83^{4} +O\left(83^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
