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:
| $r_{ 1 }$ |
$=$ |
$ 55 + 45\cdot 101 + 92\cdot 101^{2} + 32\cdot 101^{3} + 36\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 66 a + 93 + \left(63 a + 84\right)\cdot 101 + \left(26 a + 83\right)\cdot 101^{2} + \left(32 a + 83\right)\cdot 101^{3} + \left(92 a + 65\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 27\cdot 101 + 96\cdot 101^{2} + 44\cdot 101^{3} + 22\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 a + 6 + \left(56 a + 10\right)\cdot 101 + \left(17 a + 46\right)\cdot 101^{2} + \left(76 a + 86\right)\cdot 101^{3} + \left(59 a + 58\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 54 + \left(37 a + 71\right)\cdot 101 + \left(74 a + 25\right)\cdot 101^{2} + \left(68 a + 85\right)\cdot 101^{3} + \left(8 a + 99\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 92 + \left(44 a + 63\right)\cdot 101 + \left(83 a + 59\right)\cdot 101^{2} + \left(24 a + 70\right)\cdot 101^{3} + \left(41 a + 19\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $-2$ |
| $4$ | $3$ | $(3,4,6)$ | $1$ |
| $18$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,5,6)$ | $0$ |
| $12$ | $6$ | $(1,2)(3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
