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:
| $r_{ 1 }$ |
$=$ |
$ 58 a + 85 + \left(40 a + 39\right)\cdot 101 + \left(84 a + 56\right)\cdot 101^{2} + \left(53 a + 56\right)\cdot 101^{3} + \left(85 a + 85\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 62\cdot 101 + 24\cdot 101^{2} + 85\cdot 101^{3} + 44\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 14 + \left(60 a + 43\right)\cdot 101 + \left(16 a + 50\right)\cdot 101^{2} + \left(47 a + 86\right)\cdot 101^{3} + \left(15 a + 70\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 35\cdot 101 + 30\cdot 101^{2} + 57\cdot 101^{3} + 22\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 a + 54 + \left(95 a + 4\right)\cdot 101 + \left(18 a + 30\right)\cdot 101^{2} + \left(40 a + 89\right)\cdot 101^{3} + \left(51 a + 57\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 19 + \left(5 a + 17\right)\cdot 101 + \left(82 a + 10\right)\cdot 101^{2} + \left(60 a + 29\right)\cdot 101^{3} + \left(49 a + 21\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5,6,4,2)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $20$ | $3$ | $(1,5,4)(2,3,6)$ | $1$ |
| $30$ | $4$ | $(1,3,2,5)$ | $0$ |
| $24$ | $5$ | $(1,4,3,5,6)$ | $-1$ |
| $20$ | $6$ | $(1,3,5,6,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
