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:
| $r_{ 1 }$ |
$=$ |
$ 84 + 97\cdot 103 + 96\cdot 103^{2} + 49\cdot 103^{3} + 65\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 22 + \left(24 a + 8\right)\cdot 103 + \left(61 a + 60\right)\cdot 103^{2} + \left(71 a + 98\right)\cdot 103^{3} + \left(72 a + 72\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 35\cdot 103 + 53\cdot 103^{2} + 8\cdot 103^{3} + 42\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 94 a + 7 + \left(49 a + 5\right)\cdot 103 + \left(47 a + 2\right)\cdot 103^{2} + \left(11 a + 91\right)\cdot 103^{3} + \left(79 a + 44\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 a + 59 + \left(78 a + 98\right)\cdot 103 + \left(41 a + 96\right)\cdot 103^{2} + \left(31 a + 5\right)\cdot 103^{3} + \left(30 a + 74\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 101 + \left(53 a + 63\right)\cdot 103 + \left(55 a + 102\right)\cdot 103^{2} + \left(91 a + 54\right)\cdot 103^{3} + \left(23 a + 9\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
