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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 71 + \left(35 a + 53\right)\cdot 109 + \left(46 a + 74\right)\cdot 109^{2} + \left(54 a + 75\right)\cdot 109^{3} + \left(67 a + 81\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 57 + \left(58 a + 81\right)\cdot 109 + \left(48 a + 17\right)\cdot 109^{2} + \left(81 a + 68\right)\cdot 109^{3} + \left(85 a + 99\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 86\cdot 109 + 42\cdot 109^{2} + 93\cdot 109^{3} + 105\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 102 a + 78 + \left(73 a + 81\right)\cdot 109 + \left(62 a + 85\right)\cdot 109^{2} + \left(54 a + 83\right)\cdot 109^{3} + \left(41 a + 94\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 39 + \left(79 a + 39\right)\cdot 109 + \left(44 a + 66\right)\cdot 109^{2} + \left(44 a + 61\right)\cdot 109^{3} + \left(70 a + 16\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a + 84 + \left(29 a + 73\right)\cdot 109 + \left(64 a + 31\right)\cdot 109^{2} + \left(64 a + 61\right)\cdot 109^{3} + \left(38 a + 42\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 a + 68 + \left(50 a + 19\right)\cdot 109 + \left(60 a + 8\right)\cdot 109^{2} + \left(27 a + 101\right)\cdot 109^{3} + \left(23 a + 103\right)\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.
