Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 30 + \left(45 a + 8\right)\cdot 107 + \left(76 a + 28\right)\cdot 107^{2} + \left(52 a + 92\right)\cdot 107^{3} + \left(101 a + 29\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 63 + \left(45 a + 88\right)\cdot 107 + \left(76 a + 92\right)\cdot 107^{2} + \left(52 a + 96\right)\cdot 107^{3} + \left(101 a + 2\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 83 + \left(45 a + 14\right)\cdot 107 + \left(76 a + 77\right)\cdot 107^{2} + \left(52 a + 90\right)\cdot 107^{3} + \left(101 a + 25\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 a + 37 + \left(61 a + 47\right)\cdot 107 + \left(30 a + 16\right)\cdot 107^{2} + \left(54 a + 11\right)\cdot 107^{3} + \left(5 a + 58\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 a + 17 + \left(61 a + 14\right)\cdot 107 + \left(30 a + 32\right)\cdot 107^{2} + \left(54 a + 17\right)\cdot 107^{3} + \left(5 a + 35\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 a + 91 + \left(61 a + 40\right)\cdot 107 + \left(30 a + 74\right)\cdot 107^{2} + \left(54 a + 12\right)\cdot 107^{3} + \left(5 a + 62\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,5,3,6,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,3,6,2,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,4,2,6,3,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
