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:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 71 + \left(75 a + 82\right)\cdot 83 + \left(23 a + 70\right)\cdot 83^{2} + \left(28 a + 32\right)\cdot 83^{3} + \left(7 a + 37\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 67\cdot 83 + 82\cdot 83^{2} + 33\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 a + 71 + \left(37 a + 64\right)\cdot 83 + \left(58 a + 26\right)\cdot 83^{2} + \left(67 a + 33\right)\cdot 83^{3} + \left(a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 a + 27 + \left(11 a + 33\right)\cdot 83 + \left(27 a + 34\right)\cdot 83^{2} + \left(33 a + 31\right)\cdot 83^{3} + \left(56 a + 8\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 70 + \left(45 a + 20\right)\cdot 83 + \left(24 a + 47\right)\cdot 83^{2} + \left(15 a + 42\right)\cdot 83^{3} + \left(81 a + 23\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 a + 80 + \left(7 a + 65\right)\cdot 83 + \left(59 a + 19\right)\cdot 83^{2} + \left(54 a + 37\right)\cdot 83^{3} + \left(75 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 a + 75 + \left(71 a + 79\right)\cdot 83 + \left(55 a + 49\right)\cdot 83^{2} + \left(49 a + 37\right)\cdot 83^{3} + \left(26 a + 31\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 value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
