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:
| $r_{ 1 }$ |
$=$ |
$ 102 a + 28 + \left(58 a + 68\right)\cdot 107 + \left(34 a + 64\right)\cdot 107^{2} + \left(4 a + 1\right)\cdot 107^{3} + \left(96 a + 6\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 8 + \left(48 a + 95\right)\cdot 107 + \left(72 a + 36\right)\cdot 107^{2} + \left(102 a + 91\right)\cdot 107^{3} + \left(10 a + 64\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 93 a + 44 + \left(88 a + 55\right)\cdot 107 + \left(45 a + 27\right)\cdot 107^{2} + \left(83 a + 19\right)\cdot 107^{3} + \left(22 a + 59\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + 45 + \left(77 a + 30\right)\cdot 107 + \left(84 a + 69\right)\cdot 107^{2} + \left(67 a + 64\right)\cdot 107^{3} + \left(10 a + 87\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 95 + \left(18 a + 103\right)\cdot 107 + \left(61 a + 14\right)\cdot 107^{2} + \left(23 a + 93\right)\cdot 107^{3} + \left(84 a + 66\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 5\cdot 107 + 98\cdot 107^{2} + 13\cdot 107^{3} + 81\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 a + 59 + \left(29 a + 69\right)\cdot 107 + \left(22 a + 9\right)\cdot 107^{2} + \left(39 a + 37\right)\cdot 107^{3} + \left(96 a + 62\right)\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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
