Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 79 a + 25 + \left(82 a + 67\right)\cdot 103 + \left(24 a + 4\right)\cdot 103^{2} + \left(47 a + 4\right)\cdot 103^{3} + \left(62 a + 30\right)\cdot 103^{4} + \left(92 a + 30\right)\cdot 103^{5} + \left(67 a + 94\right)\cdot 103^{6} + \left(2 a + 65\right)\cdot 103^{7} + \left(33 a + 30\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 1 + \left(20 a + 71\right)\cdot 103 + \left(78 a + 49\right)\cdot 103^{2} + \left(55 a + 26\right)\cdot 103^{3} + \left(40 a + 45\right)\cdot 103^{4} + \left(10 a + 60\right)\cdot 103^{5} + \left(35 a + 69\right)\cdot 103^{6} + 100 a\cdot 103^{7} + \left(69 a + 61\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 70 + \left(87 a + 67\right)\cdot 103 + \left(a + 94\right)\cdot 103^{2} + \left(53 a + 19\right)\cdot 103^{3} + \left(53 a + 42\right)\cdot 103^{4} + \left(98 a + 37\right)\cdot 103^{5} + \left(17 a + 43\right)\cdot 103^{6} + \left(a + 78\right)\cdot 103^{7} + \left(96 a + 23\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 90 a + 83 + \left(15 a + 38\right)\cdot 103 + \left(101 a + 9\right)\cdot 103^{2} + \left(49 a + 71\right)\cdot 103^{3} + \left(49 a + 42\right)\cdot 103^{4} + \left(4 a + 82\right)\cdot 103^{5} + \left(85 a + 65\right)\cdot 103^{6} + \left(101 a + 61\right)\cdot 103^{7} + \left(6 a + 15\right)\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 + 52\cdot 103 + 28\cdot 103^{2} + 46\cdot 103^{3} + 92\cdot 103^{4} + 19\cdot 103^{5} + 23\cdot 103^{6} + 70\cdot 103^{7} + 21\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 11\cdot 103 + 19\cdot 103^{2} + 38\cdot 103^{3} + 56\cdot 103^{4} + 78\cdot 103^{5} + 12\cdot 103^{6} + 32\cdot 103^{7} + 53\cdot 103^{8} +O\left(103^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(1,5,3)(2,6,4)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $3$ |
$2$ |
$(3,4)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,2,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,3,2,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
