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:
| $r_{ 1 }$ |
$=$ |
$ 44 a + 15 + \left(29 a + 49\right)\cdot 109 + \left(99 a + 77\right)\cdot 109^{2} + \left(82 a + 15\right)\cdot 109^{3} + \left(104 a + 68\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 83\cdot 109 + 63\cdot 109^{2} + 76\cdot 109^{3} + 36\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 + 109 + 30\cdot 109^{2} + 8\cdot 109^{3} + 103\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 70 a + 17 + \left(88 a + 15\right)\cdot 109 + \left(50 a + 23\right)\cdot 109^{2} + \left(90 a + 39\right)\cdot 109^{3} + \left(70 a + 24\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 87 + \left(20 a + 33\right)\cdot 109 + \left(58 a + 94\right)\cdot 109^{2} + \left(18 a + 78\right)\cdot 109^{3} + \left(38 a + 4\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 a + 59 + \left(79 a + 34\right)\cdot 109 + \left(9 a + 38\right)\cdot 109^{2} + \left(26 a + 108\right)\cdot 109^{3} + \left(4 a + 89\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,4)(3,5)$ |
| $(1,3,2,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,4)(3,5)$ | $2$ |
| $15$ | $2$ | $(1,6)(2,3)$ | $0$ |
| $20$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $30$ | $4$ | $(1,3,6,2)$ | $0$ |
| $24$ | $5$ | $(1,4,3,2,5)$ | $-1$ |
| $20$ | $6$ | $(1,3,2,5,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
