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:
| $r_{ 1 }$ |
$=$ |
$ 81 a + 20 + \left(81 a + 80\right)\cdot 83 + \left(18 a + 17\right)\cdot 83^{2} + \left(44 a + 62\right)\cdot 83^{3} + \left(69 a + 17\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 40 + \left(41 a + 10\right)\cdot 83 + \left(43 a + 69\right)\cdot 83^{2} + \left(3 a + 66\right)\cdot 83^{3} + \left(14 a + 64\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 30 + \left(21 a + 23\right)\cdot 83 + \left(2 a + 43\right)\cdot 83^{2} + \left(29 a + 8\right)\cdot 83^{3} + \left(70 a + 62\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 18 + \left(a + 81\right)\cdot 83 + \left(64 a + 37\right)\cdot 83^{2} + \left(38 a + 4\right)\cdot 83^{3} + \left(13 a + 43\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 57 + \left(61 a + 17\right)\cdot 83 + \left(80 a + 24\right)\cdot 83^{2} + \left(53 a + 35\right)\cdot 83^{3} + \left(12 a + 20\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 a + 42 + \left(41 a + 49\right)\cdot 83 + \left(39 a + 71\right)\cdot 83^{2} + \left(79 a + 26\right)\cdot 83^{3} + \left(68 a + 75\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 + 69\cdot 83 + 67\cdot 83^{2} + 44\cdot 83^{3} + 48\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.
