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:
| $r_{ 1 }$ |
$=$ |
$ 63 a + 75 + \left(18 a + 56\right)\cdot 103 + \left(15 a + 2\right)\cdot 103^{2} + \left(79 a + 90\right)\cdot 103^{3} + \left(40 a + 96\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 a + 55 + \left(19 a + 29\right)\cdot 103 + \left(34 a + 87\right)\cdot 103^{2} + \left(32 a + 48\right)\cdot 103^{3} + \left(66 a + 3\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 62\cdot 103 + 57\cdot 103^{2} + 58\cdot 103^{3} + 72\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 a + 85 + \left(49 a + 96\right)\cdot 103 + \left(48 a + 30\right)\cdot 103^{2} + \left(93 a + 87\right)\cdot 103^{3} + \left(89 a + 21\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 48 + \left(53 a + 80\right)\cdot 103 + \left(54 a + 29\right)\cdot 103^{2} + \left(9 a + 29\right)\cdot 103^{3} + \left(13 a + 18\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 35 + \left(84 a + 12\right)\cdot 103 + \left(87 a + 102\right)\cdot 103^{2} + \left(23 a + 50\right)\cdot 103^{3} + \left(62 a + 58\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 a + 31 + \left(83 a + 73\right)\cdot 103 + \left(68 a + 101\right)\cdot 103^{2} + \left(70 a + 46\right)\cdot 103^{3} + \left(36 a + 37\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.
