Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 87 + \left(8 a + 4\right)\cdot 101 + \left(98 a + 42\right)\cdot 101^{2} + \left(85 a + 35\right)\cdot 101^{3} + \left(71 a + 15\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 84 a + 78 + \left(73 a + 87\right)\cdot 101 + \left(99 a + 77\right)\cdot 101^{2} + \left(22 a + 7\right)\cdot 101^{3} + \left(22 a + 3\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 a + 41 + \left(92 a + 100\right)\cdot 101 + \left(2 a + 21\right)\cdot 101^{2} + \left(15 a + 79\right)\cdot 101^{3} + \left(29 a + 14\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 10 + \left(27 a + 97\right)\cdot 101 + \left(a + 99\right)\cdot 101^{2} + \left(78 a + 100\right)\cdot 101^{3} + \left(78 a + 68\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 96\cdot 101 + 36\cdot 101^{2} + 87\cdot 101^{3} + 70\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 17\cdot 101 + 24\cdot 101^{2} + 93\cdot 101^{3} + 28\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
| $4$ | $3$ | $(1,3,5)$ | $-2$ |
| $18$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $1$ |
| $12$ | $6$ | $(1,3,5)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
