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:
| $r_{ 1 }$ |
$=$ |
$ 74 a + 48 + \left(20 a + 6\right)\cdot 101 + \left(26 a + 45\right)\cdot 101^{2} + \left(73 a + 43\right)\cdot 101^{3} + \left(6 a + 21\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 41 + \left(80 a + 15\right)\cdot 101 + \left(74 a + 28\right)\cdot 101^{2} + \left(27 a + 7\right)\cdot 101^{3} + \left(94 a + 76\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 a + 72 + \left(33 a + 14\right)\cdot 101 + \left(40 a + 32\right)\cdot 101^{2} + \left(27 a + 33\right)\cdot 101^{3} + \left(82 a + 60\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 49\cdot 101 + 64\cdot 101^{2} + 60\cdot 101^{3} + 54\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a + 70 + \left(67 a + 98\right)\cdot 101 + \left(60 a + 58\right)\cdot 101^{2} + \left(73 a + 1\right)\cdot 101^{3} + \left(18 a + 59\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 17\cdot 101 + 74\cdot 101^{2} + 55\cdot 101^{3} + 31\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2,4,5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $15$ | $2$ | $(1,4)(2,5)$ | $0$ |
| $20$ | $3$ | $(1,5,2)(3,6,4)$ | $1$ |
| $30$ | $4$ | $(2,6,4,5)$ | $0$ |
| $24$ | $5$ | $(1,4,6,2,3)$ | $-1$ |
| $20$ | $6$ | $(1,6,5,4,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
