Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 57 + \left(42 a + 4\right)\cdot 107 + \left(6 a + 105\right)\cdot 107^{2} + \left(76 a + 29\right)\cdot 107^{3} + \left(40 a + 56\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 80 a + 58 + \left(64 a + 39\right)\cdot 107 + \left(100 a + 88\right)\cdot 107^{2} + \left(30 a + 6\right)\cdot 107^{3} + \left(66 a + 36\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 86 + 48\cdot 107 + 34\cdot 107^{3} + 47\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 102\cdot 107 + 87\cdot 107^{2} + 70\cdot 107^{3} + 75\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 87\cdot 107 + 92\cdot 107^{2} + 83\cdot 107^{3} + 93\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 90\cdot 107 + 76\cdot 107^{2} + 34\cdot 107^{3} + 60\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 + 55\cdot 107 + 83\cdot 107^{2} + 60\cdot 107^{3} + 58\cdot 107^{4} +O\left(107^{ 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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
