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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 43 + \left(22 a + 90\right)\cdot 107 + \left(98 a + 36\right)\cdot 107^{2} + \left(37 a + 43\right)\cdot 107^{3} + \left(30 a + 101\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 86 a + 20 + \left(84 a + 51\right)\cdot 107 + \left(8 a + 86\right)\cdot 107^{2} + \left(69 a + 96\right)\cdot 107^{3} + \left(76 a + 77\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 + 77\cdot 107 + 32\cdot 107^{2} + 29\cdot 107^{3} + 15\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 a + 83 + \left(86 a + 66\right)\cdot 107 + \left(23 a + 31\right)\cdot 107^{2} + \left(64 a + 9\right)\cdot 107^{3} + \left(57 a + 24\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 9\cdot 107 + 93\cdot 107^{2} + 6\cdot 107^{3} + 19\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 30 + \left(20 a + 25\right)\cdot 107 + \left(83 a + 40\right)\cdot 107^{2} + \left(42 a + 28\right)\cdot 107^{3} + \left(49 a + 83\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,4)(3,6)$ |
| $(1,2,6,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ |
| $15$ | $2$ | $(1,5)(2,6)$ | $0$ |
| $20$ | $3$ | $(1,6,3)(2,4,5)$ | $1$ |
| $30$ | $4$ | $(1,2,5,6)$ | $0$ |
| $24$ | $5$ | $(1,3,2,6,4)$ | $-1$ |
| $20$ | $6$ | $(1,2,6,4,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
