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:
| $r_{ 1 }$ |
$=$ |
$ 35 + 48\cdot 103 + 96\cdot 103^{2} + 25\cdot 103^{3} + 39\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 52 + \left(25 a + 35\right)\cdot 103 + \left(34 a + 85\right)\cdot 103^{2} + \left(29 a + 80\right)\cdot 103^{3} + \left(a + 33\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 a + 85 + \left(95 a + 101\right)\cdot 103 + \left(52 a + 71\right)\cdot 103^{2} + \left(53 a + 82\right)\cdot 103^{3} + \left(13 a + 67\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a + 44 + \left(7 a + 32\right)\cdot 103 + \left(50 a + 29\right)\cdot 103^{2} + \left(49 a + 83\right)\cdot 103^{3} + \left(89 a + 27\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 7 + \left(77 a + 3\right)\cdot 103 + \left(68 a + 94\right)\cdot 103^{2} + \left(73 a + 75\right)\cdot 103^{3} + \left(101 a + 5\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 88 + 87\cdot 103 + 34\cdot 103^{2} + 63\cdot 103^{3} + 31\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,5,4,6,3,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,6)$ |
$0$ |
| $20$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$1$ |
| $30$ |
$4$ |
$(1,3,2,6)$ |
$0$ |
| $24$ |
$5$ |
$(1,5,3,4,2)$ |
$-1$ |
| $20$ |
$6$ |
$(1,5,4,6,3,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
