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:
| $r_{ 1 }$ |
$=$ |
$ 106 a + 24 + \left(99 a + 79\right)\cdot 107 + \left(4 a + 61\right)\cdot 107^{2} + \left(69 a + 99\right)\cdot 107^{3} + \left(31 a + 60\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 3 + \left(96 a + 6\right)\cdot 107 + \left(58 a + 22\right)\cdot 107^{2} + \left(106 a + 91\right)\cdot 107^{3} + \left(75 a + 3\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 73\cdot 107 + 97\cdot 107^{2} + 78\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 9 + \left(10 a + 16\right)\cdot 107 + \left(48 a + 54\right)\cdot 107^{2} + 30\cdot 107^{3} + \left(31 a + 94\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 79 + 93\cdot 107 + 103\cdot 107^{2} + 48\cdot 107^{3} + 72\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 20 + \left(7 a + 52\right)\cdot 107 + \left(102 a + 88\right)\cdot 107^{2} + \left(37 a + 49\right)\cdot 107^{3} + \left(75 a + 11\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(2,3,4)$ | $-2$ |
| $18$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,2,5,3,6,4)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
