Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 83 a + 80 + \left(19 a + 76\right)\cdot 109 + \left(53 a + 64\right)\cdot 109^{2} + \left(75 a + 99\right)\cdot 109^{3} + \left(58 a + 21\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 72\cdot 109 + 101\cdot 109^{2} + 77\cdot 109^{3} + 6\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 a + 66 + \left(8 a + 83\right)\cdot 109 + \left(14 a + 91\right)\cdot 109^{2} + \left(57 a + 25\right)\cdot 109^{3} + \left(108 a + 64\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 54 + \left(89 a + 13\right)\cdot 109 + \left(55 a + 98\right)\cdot 109^{2} + \left(33 a + 12\right)\cdot 109^{3} + \left(50 a + 5\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 30\cdot 109 + 101\cdot 109^{2} + 104\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 23\cdot 109 + 99\cdot 109^{2} + 45\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 a + 23 + \left(100 a + 26\right)\cdot 109 + \left(94 a + 97\right)\cdot 109^{2} + \left(51 a + 68\right)\cdot 109^{3} + 6\cdot 109^{4} +O\left(109^{ 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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
