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:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 5 + \left(73 a + 100\right)\cdot 107 + \left(51 a + 21\right)\cdot 107^{2} + \left(73 a + 25\right)\cdot 107^{3} + \left(20 a + 21\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 24 + \left(83 a + 11\right)\cdot 107 + \left(98 a + 8\right)\cdot 107^{2} + \left(92 a + 81\right)\cdot 107^{3} + \left(63 a + 64\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 a + 54 + \left(33 a + 33\right)\cdot 107 + \left(55 a + 48\right)\cdot 107^{2} + \left(33 a + 53\right)\cdot 107^{3} + \left(86 a + 30\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 2\cdot 107 + 100\cdot 107^{2} + 99\cdot 107^{3} + 28\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 80\cdot 107 + 36\cdot 107^{2} + 28\cdot 107^{3} + 55\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 a + 65 + \left(23 a + 93\right)\cdot 107 + \left(8 a + 105\right)\cdot 107^{2} + \left(14 a + 32\right)\cdot 107^{3} + \left(43 a + 13\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $4$ | $3$ | $(2,4,6)$ | $1$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
