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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 69\cdot 103 + 4\cdot 103^{2} + 23\cdot 103^{3} + 17\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 101 a + 91 + \left(65 a + 85\right)\cdot 103 + \left(23 a + 18\right)\cdot 103^{2} + \left(20 a + 93\right)\cdot 103^{3} + \left(88 a + 8\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 87 + \left(102 a + 43\right)\cdot 103 + \left(102 a + 25\right)\cdot 103^{2} + \left(93 a + 66\right)\cdot 103^{3} + \left(31 a + 56\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 89 + \left(37 a + 50\right)\cdot 103 + \left(79 a + 79\right)\cdot 103^{2} + \left(82 a + 89\right)\cdot 103^{3} + \left(14 a + 76\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 a + 90 + 39\cdot 103 + 26\cdot 103^{2} + \left(9 a + 57\right)\cdot 103^{3} + \left(71 a + 97\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 19\cdot 103 + 51\cdot 103^{2} + 82\cdot 103^{3} + 51\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
