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:
| $r_{ 1 }$ |
$=$ |
$ 76 a + 68 + \left(39 a + 13\right)\cdot 83 + \left(57 a + 69\right)\cdot 83^{2} + \left(68 a + 71\right)\cdot 83^{3} + \left(17 a + 45\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 a + 47 + \left(39 a + 12\right)\cdot 83 + \left(57 a + 38\right)\cdot 83^{2} + \left(68 a + 2\right)\cdot 83^{3} + \left(17 a + 45\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 40 + \left(43 a + 59\right)\cdot 83 + \left(25 a + 55\right)\cdot 83^{2} + \left(14 a + 13\right)\cdot 83^{3} + \left(65 a + 77\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 55 + \left(43 a + 74\right)\cdot 83 + \left(25 a + 49\right)\cdot 83^{2} + \left(14 a + 44\right)\cdot 83^{3} + \left(65 a + 17\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 a + 62 + \left(39 a + 27\right)\cdot 83 + \left(57 a + 32\right)\cdot 83^{2} + \left(68 a + 33\right)\cdot 83^{3} + \left(17 a + 68\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 61 + \left(43 a + 60\right)\cdot 83 + \left(25 a + 3\right)\cdot 83^{2} + 14 a\cdot 83^{3} + \left(65 a + 78\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,5,6,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,2,6,5,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
