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:
| $r_{ 1 }$ |
$=$ |
$ 90 a + 50 + \left(29 a + 88\right)\cdot 101 + \left(89 a + 37\right)\cdot 101^{2} + \left(83 a + 87\right)\cdot 101^{3} + \left(36 a + 82\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 6 + \left(71 a + 17\right)\cdot 101 + \left(11 a + 62\right)\cdot 101^{2} + \left(17 a + 30\right)\cdot 101^{3} + \left(64 a + 45\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 48 + 31\cdot 101 + 100\cdot 101^{2} + 32\cdot 101^{3} + 5\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 a + 6 + \left(42 a + 84\right)\cdot 101 + \left(46 a + 80\right)\cdot 101^{2} + \left(83 a + 83\right)\cdot 101^{3} + \left(76 a + 73\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 92 + \left(58 a + 81\right)\cdot 101 + \left(54 a + 21\right)\cdot 101^{2} + \left(17 a + 68\right)\cdot 101^{3} + \left(24 a + 95\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
