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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 87 + \left(70 a + 66\right)\cdot 101 + \left(85 a + 17\right)\cdot 101^{2} + \left(22 a + 89\right)\cdot 101^{3} + \left(50 a + 4\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 73 a + 10 + \left(72 a + 92\right)\cdot 101 + \left(12 a + 28\right)\cdot 101^{2} + \left(96 a + 60\right)\cdot 101^{3} + \left(83 a + 21\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 101 + 10\cdot 101^{2} + 15\cdot 101^{3} + 49\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 100 + \left(28 a + 6\right)\cdot 101 + \left(88 a + 7\right)\cdot 101^{2} + \left(4 a + 28\right)\cdot 101^{3} + \left(17 a + 59\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 89 a + 34 + \left(30 a + 32\right)\cdot 101 + \left(15 a + 88\right)\cdot 101^{2} + \left(78 a + 94\right)\cdot 101^{3} + \left(50 a + 81\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 2\cdot 101 + 50\cdot 101^{2} + 15\cdot 101^{3} + 86\cdot 101^{4} +O\left(101^{ 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.
