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:
| $r_{ 1 }$ |
$=$ |
$ 105 a + 4 + \left(13 a + 78\right)\cdot 107 + \left(79 a + 62\right)\cdot 107^{2} + \left(11 a + 69\right)\cdot 107^{3} + \left(70 a + 79\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 a + 21 + \left(54 a + 73\right)\cdot 107 + \left(101 a + 91\right)\cdot 107^{2} + \left(56 a + 43\right)\cdot 107^{3} + \left(31 a + 72\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 78 a + 58 + \left(45 a + 54\right)\cdot 107 + \left(35 a + 5\right)\cdot 107^{2} + \left(49 a + 26\right)\cdot 107^{3} + \left(85 a + 14\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 103 + \left(93 a + 28\right)\cdot 107 + \left(27 a + 44\right)\cdot 107^{2} + \left(95 a + 37\right)\cdot 107^{3} + \left(36 a + 27\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 a + 86 + \left(52 a + 33\right)\cdot 107 + \left(5 a + 15\right)\cdot 107^{2} + \left(50 a + 63\right)\cdot 107^{3} + \left(75 a + 34\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 49 + \left(61 a + 52\right)\cdot 107 + \left(71 a + 101\right)\cdot 107^{2} + \left(57 a + 80\right)\cdot 107^{3} + \left(21 a + 92\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
