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:
| $r_{ 1 }$ |
$=$ |
$ 86 a + 30 + \left(69 a + 80\right)\cdot 103 + \left(31 a + 34\right)\cdot 103^{2} + \left(11 a + 19\right)\cdot 103^{3} + \left(85 a + 30\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 98 a + 78 + \left(43 a + 60\right)\cdot 103 + \left(64 a + 29\right)\cdot 103^{2} + \left(94 a + 68\right)\cdot 103^{3} + \left(81 a + 98\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 13 + \left(33 a + 64\right)\cdot 103 + \left(71 a + 99\right)\cdot 103^{2} + \left(91 a + 101\right)\cdot 103^{3} + 17 a\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 a + 83 + \left(17 a + 81\right)\cdot 103 + \left(85 a + 22\right)\cdot 103^{2} + \left(37 a + 64\right)\cdot 103^{3} + \left(66 a + 40\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 66 + \left(85 a + 13\right)\cdot 103 + \left(17 a + 90\right)\cdot 103^{2} + \left(65 a + 16\right)\cdot 103^{3} + \left(36 a + 69\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 103 + 85\cdot 103^{2} + 42\cdot 103^{3} + 86\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a + 73 + \left(59 a + 6\right)\cdot 103 + \left(38 a + 50\right)\cdot 103^{2} + \left(8 a + 98\right)\cdot 103^{3} + \left(21 a + 85\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
