Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 + 69\cdot 101 + 101^{2} + 90\cdot 101^{3} + 80\cdot 101^{4} + 13\cdot 101^{5} + 95\cdot 101^{6} + 43\cdot 101^{7} + 7\cdot 101^{8} + 99\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 89 + \left(94 a + 34\right)\cdot 101 + \left(37 a + 23\right)\cdot 101^{2} + \left(91 a + 27\right)\cdot 101^{3} + \left(88 a + 10\right)\cdot 101^{4} + \left(77 a + 96\right)\cdot 101^{5} + \left(100 a + 60\right)\cdot 101^{6} + \left(15 a + 47\right)\cdot 101^{7} + \left(26 a + 63\right)\cdot 101^{8} + \left(92 a + 15\right)\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 a + 27 + \left(6 a + 74\right)\cdot 101 + \left(63 a + 80\right)\cdot 101^{2} + \left(9 a + 51\right)\cdot 101^{3} + \left(12 a + 72\right)\cdot 101^{4} + \left(23 a + 15\right)\cdot 101^{5} + 83\cdot 101^{6} + \left(85 a + 10\right)\cdot 101^{7} + \left(74 a + 51\right)\cdot 101^{8} + \left(8 a + 55\right)\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 56\cdot 101 + 65\cdot 101^{2} + 16\cdot 101^{3} + 18\cdot 101^{4} + 69\cdot 101^{5} + 10\cdot 101^{6} + 43\cdot 101^{7} + 31\cdot 101^{8} + 76\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 27 + \left(61 a + 21\right)\cdot 101 + \left(a + 93\right)\cdot 101^{2} + \left(48 a + 13\right)\cdot 101^{3} + \left(61 a + 12\right)\cdot 101^{4} + \left(50 a + 34\right)\cdot 101^{5} + \left(10 a + 81\right)\cdot 101^{6} + \left(72 a + 40\right)\cdot 101^{7} + \left(29 a + 51\right)\cdot 101^{8} + \left(92 a + 60\right)\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 84 a + 95 + \left(39 a + 46\right)\cdot 101 + \left(99 a + 38\right)\cdot 101^{2} + \left(52 a + 2\right)\cdot 101^{3} + \left(39 a + 8\right)\cdot 101^{4} + \left(50 a + 74\right)\cdot 101^{5} + \left(90 a + 72\right)\cdot 101^{6} + \left(28 a + 15\right)\cdot 101^{7} + \left(71 a + 98\right)\cdot 101^{8} + \left(8 a + 96\right)\cdot 101^{9} +O\left(101^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(4,6)$ |
| $(1,2,5)(3,6,4)$ |
| $(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(5,6)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,6,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,5)(2,3)$ | $1$ |
| $8$ | $6$ | $(1,6,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
