Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 94 + \left(71 a + 54\right)\cdot 103 + \left(4 a + 74\right)\cdot 103^{2} + \left(63 a + 24\right)\cdot 103^{3} + \left(37 a + 19\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 81\cdot 103 + 57\cdot 103^{2} + 24\cdot 103^{3} + 57\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 28\cdot 103 + 20\cdot 103^{2} + 98\cdot 103^{3} + 89\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 43 + \left(75 a + 50\right)\cdot 103 + \left(92 a + 65\right)\cdot 103^{2} + \left(93 a + 38\right)\cdot 103^{3} + \left(38 a + 50\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 a + 97 + \left(31 a + 19\right)\cdot 103 + \left(98 a + 8\right)\cdot 103^{2} + \left(39 a + 83\right)\cdot 103^{3} + \left(65 a + 96\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 95 + \left(27 a + 73\right)\cdot 103 + \left(10 a + 82\right)\cdot 103^{2} + \left(9 a + 39\right)\cdot 103^{3} + \left(64 a + 98\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
