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:
| $r_{ 1 }$ |
$=$ |
$ 82 a + 52 + \left(65 a + 40\right)\cdot 103 + \left(81 a + 76\right)\cdot 103^{2} + \left(64 a + 23\right)\cdot 103^{3} + \left(65 a + 8\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 80 + \left(37 a + 24\right)\cdot 103 + \left(21 a + 33\right)\cdot 103^{2} + \left(38 a + 26\right)\cdot 103^{3} + \left(37 a + 85\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 a + 101 + \left(65 a + 40\right)\cdot 103 + \left(81 a + 17\right)\cdot 103^{2} + \left(64 a + 43\right)\cdot 103^{3} + \left(65 a + 84\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 a + 85 + \left(65 a + 45\right)\cdot 103 + \left(81 a + 88\right)\cdot 103^{2} + \left(64 a + 9\right)\cdot 103^{3} + \left(65 a + 9\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 31 + \left(37 a + 24\right)\cdot 103 + \left(21 a + 92\right)\cdot 103^{2} + \left(38 a + 6\right)\cdot 103^{3} + \left(37 a + 9\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 64 + \left(37 a + 29\right)\cdot 103 + \left(21 a + 1\right)\cdot 103^{2} + \left(38 a + 96\right)\cdot 103^{3} + \left(37 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 |
| $(1,6,3,5,4,2)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,5,4,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,4,5,3,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
