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:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 101 + \left(50 a + 31\right)\cdot 103 + \left(76 a + 27\right)\cdot 103^{2} + \left(51 a + 63\right)\cdot 103^{3} + \left(29 a + 69\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 37 + \left(52 a + 43\right)\cdot 103 + \left(26 a + 53\right)\cdot 103^{2} + \left(51 a + 38\right)\cdot 103^{3} + \left(73 a + 47\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 98 a + 27 + \left(51 a + 56\right)\cdot 103 + \left(11 a + 87\right)\cdot 103^{2} + \left(28 a + 31\right)\cdot 103^{3} + \left(59 a + 101\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 100 + \left(53 a + 69\right)\cdot 103 + \left(99 a + 23\right)\cdot 103^{2} + \left(66 a + 28\right)\cdot 103^{3} + \left(44 a + 93\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 22 + \left(51 a + 10\right)\cdot 103 + \left(91 a + 47\right)\cdot 103^{2} + \left(74 a + 48\right)\cdot 103^{3} + \left(43 a + 29\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 a + 23 + \left(49 a + 97\right)\cdot 103 + \left(3 a + 69\right)\cdot 103^{2} + \left(36 a + 98\right)\cdot 103^{3} + \left(58 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,3,4,2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,4,2,5,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,2,4,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
