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:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 63 + \left(51 a + 64\right)\cdot 103 + \left(41 a + 64\right)\cdot 103^{2} + \left(7 a + 39\right)\cdot 103^{3} + \left(22 a + 75\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 92 + 67\cdot 103 + 90\cdot 103^{2} + 67\cdot 103^{3} + 23\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 100 a + 66 + \left(51 a + 9\right)\cdot 103 + \left(61 a + 55\right)\cdot 103^{2} + \left(95 a + 5\right)\cdot 103^{3} + \left(80 a + 90\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 25\cdot 103 + 11\cdot 103^{2} + 92\cdot 103^{3} + 23\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 a + 85 + \left(20 a + 97\right)\cdot 103 + \left(87 a + 61\right)\cdot 103^{2} + \left(98 a + 97\right)\cdot 103^{3} + \left(58 a + 67\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 57 + \left(82 a + 43\right)\cdot 103 + \left(15 a + 25\right)\cdot 103^{2} + \left(4 a + 6\right)\cdot 103^{3} + \left(44 a + 28\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $6$ | $2$ | $(1,2)$ | $0$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $4$ | $3$ | $(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,3,6)$ | $1$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
