Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 a + 23 + \left(55 a + 5\right)\cdot 59 + \left(39 a + 43\right)\cdot 59^{2} + \left(41 a + 29\right)\cdot 59^{3} + \left(6 a + 15\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 31 + \left(55 a + 32\right)\cdot 59 + \left(39 a + 24\right)\cdot 59^{2} + \left(41 a + 58\right)\cdot 59^{3} + \left(6 a + 8\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 17 + \left(3 a + 43\right)\cdot 59 + \left(19 a + 8\right)\cdot 59^{2} + \left(17 a + 1\right)\cdot 59^{3} + \left(52 a + 33\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 9 + \left(3 a + 16\right)\cdot 59 + \left(19 a + 27\right)\cdot 59^{2} + \left(17 a + 31\right)\cdot 59^{3} + \left(52 a + 39\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 57 + \left(55 a + 4\right)\cdot 59 + \left(39 a + 15\right)\cdot 59^{2} + \left(41 a + 27\right)\cdot 59^{3} + \left(6 a + 57\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 43 + \left(3 a + 15\right)\cdot 59 + \left(19 a + 58\right)\cdot 59^{2} + \left(17 a + 28\right)\cdot 59^{3} + \left(52 a + 22\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,5)(3,6,4)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,5,2)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,3,5,4,2,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,2,4,5,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
