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:
| $r_{ 1 }$ |
$=$ |
$ 96 a + 90 + \left(36 a + 94\right)\cdot 107 + \left(46 a + 45\right)\cdot 107^{2} + \left(105 a + 70\right)\cdot 107^{3} + \left(5 a + 25\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 81\cdot 107 + 41\cdot 107^{2} + 104\cdot 107^{3} + 3\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 a + 58 + \left(39 a + 48\right)\cdot 107 + \left(71 a + 22\right)\cdot 107^{2} + \left(11 a + 71\right)\cdot 107^{3} + \left(85 a + 22\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 28\cdot 107 + 69\cdot 107^{2} + 10\cdot 107^{3} + 80\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 46 + \left(70 a + 39\right)\cdot 107 + \left(60 a + 87\right)\cdot 107^{2} + \left(a + 17\right)\cdot 107^{3} + \left(101 a + 51\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 25 + \left(67 a + 28\right)\cdot 107 + \left(35 a + 54\right)\cdot 107^{2} + \left(95 a + 46\right)\cdot 107^{3} + \left(21 a + 30\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
