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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 22\cdot 83 + 22\cdot 83^{2} + 70\cdot 83^{3} + 8\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 54\cdot 83 + 66\cdot 83^{2} + 19\cdot 83^{3} + 75\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 10\cdot 83^{2} + 59\cdot 83^{3} + 7\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 37 + \left(78 a + 41\right)\cdot 83 + \left(47 a + 58\right)\cdot 83^{2} + \left(70 a + 28\right)\cdot 83^{3} + \left(58 a + 81\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 a + 72 + \left(43 a + 73\right)\cdot 83 + \left(14 a + 4\right)\cdot 83^{2} + \left(25 a + 46\right)\cdot 83^{3} + \left(19 a + 47\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 71 + \left(39 a + 35\right)\cdot 83 + \left(68 a + 58\right)\cdot 83^{2} + \left(57 a + 56\right)\cdot 83^{3} + \left(63 a + 41\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 a + 53 + \left(4 a + 20\right)\cdot 83 + \left(35 a + 28\right)\cdot 83^{2} + \left(12 a + 51\right)\cdot 83^{3} + \left(24 a + 69\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
