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:
| $r_{ 1 }$ |
$=$ |
$ 95 a + 44 + \left(53 a + 46\right)\cdot 109 + \left(62 a + 63\right)\cdot 109^{2} + \left(26 a + 76\right)\cdot 109^{3} + \left(37 a + 96\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 30 + \left(55 a + 5\right)\cdot 109 + \left(46 a + 72\right)\cdot 109^{2} + \left(82 a + 40\right)\cdot 109^{3} + \left(71 a + 107\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 3\cdot 109 + 35\cdot 109^{2} + 36\cdot 109^{3} + 95\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 21\cdot 109 + 68\cdot 109^{2} + 81\cdot 109^{3} + 65\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 56 + \left(97 a + 51\right)\cdot 109 + \left(a + 37\right)\cdot 109^{2} + \left(64 a + 69\right)\cdot 109^{3} + \left(17 a + 58\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a + 6 + \left(11 a + 90\right)\cdot 109 + \left(107 a + 50\right)\cdot 109^{2} + \left(44 a + 22\right)\cdot 109^{3} + \left(91 a + 12\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,3,4,5,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $30$ | $4$ | $(1,3,2,4)$ | $0$ |
| $24$ | $5$ | $(1,5,6,2,4)$ | $-1$ |
| $20$ | $6$ | $(1,3,4,5,6,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
