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:
| $r_{ 1 }$ |
$=$ |
$ 7 + 25\cdot 103 + 98\cdot 103^{2} + 36\cdot 103^{3} + 98\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 a + 69 + \left(66 a + 68\right)\cdot 103 + \left(48 a + 90\right)\cdot 103^{2} + \left(44 a + 93\right)\cdot 103^{3} + \left(16 a + 33\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 81 + \left(57 a + 5\right)\cdot 103 + \left(43 a + 70\right)\cdot 103^{2} + \left(74 a + 19\right)\cdot 103^{3} + \left(6 a + 80\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 1 + \left(30 a + 42\right)\cdot 103 + \left(96 a + 30\right)\cdot 103^{2} + 5\cdot 103^{3} + \left(53 a + 13\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 44 + \left(36 a + 57\right)\cdot 103 + \left(54 a + 72\right)\cdot 103^{2} + \left(58 a + 89\right)\cdot 103^{3} + \left(86 a + 5\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 a + 22 + \left(72 a + 51\right)\cdot 103 + \left(6 a + 96\right)\cdot 103^{2} + \left(102 a + 12\right)\cdot 103^{3} + \left(49 a + 65\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 a + 85 + \left(45 a + 58\right)\cdot 103 + \left(59 a + 56\right)\cdot 103^{2} + \left(28 a + 50\right)\cdot 103^{3} + \left(96 a + 12\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$20$ |
| $21$ |
$2$ |
$(1,2)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-4$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
