Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 5 + \left(40 a + 56\right)\cdot 101 + \left(95 a + 32\right)\cdot 101^{2} + \left(31 a + 15\right)\cdot 101^{3} + \left(81 a + 73\right)\cdot 101^{4} + \left(54 a + 1\right)\cdot 101^{5} + \left(38 a + 58\right)\cdot 101^{6} + \left(59 a + 17\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 2\cdot 101 + 79\cdot 101^{2} + 28\cdot 101^{3} + 43\cdot 101^{4} + 87\cdot 101^{5} + 24\cdot 101^{6} + 27\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 a + 87 + \left(7 a + 13\right)\cdot 101 + \left(64 a + 26\right)\cdot 101^{2} + \left(74 a + 53\right)\cdot 101^{3} + \left(91 a + 88\right)\cdot 101^{4} + \left(44 a + 36\right)\cdot 101^{5} + \left(26 a + 13\right)\cdot 101^{6} + \left(76 a + 97\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 11 + \left(93 a + 64\right)\cdot 101 + \left(36 a + 72\right)\cdot 101^{2} + \left(26 a + 85\right)\cdot 101^{3} + \left(9 a + 77\right)\cdot 101^{4} + \left(56 a + 23\right)\cdot 101^{5} + \left(74 a + 74\right)\cdot 101^{6} + \left(24 a + 72\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 a + 100 + \left(60 a + 67\right)\cdot 101 + \left(5 a + 70\right)\cdot 101^{2} + \left(69 a + 47\right)\cdot 101^{3} + \left(19 a + 63\right)\cdot 101^{4} + \left(46 a + 38\right)\cdot 101^{5} + \left(62 a + 56\right)\cdot 101^{6} + \left(41 a + 14\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 41 + \left(72 a + 11\right)\cdot 101 + \left(47 a + 19\right)\cdot 101^{2} + \left(77 a + 34\right)\cdot 101^{3} + \left(96 a + 19\right)\cdot 101^{4} + \left(78 a + 39\right)\cdot 101^{5} + \left(86 a + 32\right)\cdot 101^{6} + \left(60 a + 57\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 100\cdot 101 + 66\cdot 101^{2} + 44\cdot 101^{3} + 12\cdot 101^{4} + 19\cdot 101^{5} + 46\cdot 101^{6} + 4\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 91 a + 81 + \left(28 a + 87\right)\cdot 101 + \left(53 a + 36\right)\cdot 101^{2} + \left(23 a + 94\right)\cdot 101^{3} + \left(4 a + 25\right)\cdot 101^{4} + \left(22 a + 56\right)\cdot 101^{5} + \left(14 a + 98\right)\cdot 101^{6} + \left(40 a + 11\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(3,5)$ |
| $(1,6,3,7,4,8,5,2)$ |
| $(1,5)(2,6)(3,4)(7,8)$ |
| $(2,7)(6,8)$ |
| $(2,7)(3,5)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,5)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(2,7)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,5)(3,7)(4,6)$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(3,5)$ |
$0$ |
| $8$ |
$2$ |
$(2,6)(3,5)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,5)(2,6,7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,4,3)(2,6,7,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,4,7)(3,6,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,3)$ |
$2$ |
| $4$ |
$4$ |
$(1,5,4,3)(2,7)(6,8)$ |
$-2$ |
| $4$ |
$4$ |
$(1,6,4,8)(2,5,7,3)$ |
$0$ |
| $8$ |
$8$ |
$(1,6,3,7,4,8,5,2)$ |
$0$ |
| $8$ |
$8$ |
$(1,2,5,6,4,7,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
