Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 59 + \left(44 a + 82\right)\cdot 101 + \left(31 a + 64\right)\cdot 101^{2} + \left(39 a + 95\right)\cdot 101^{3} + \left(98 a + 80\right)\cdot 101^{4} + \left(88 a + 72\right)\cdot 101^{5} + \left(40 a + 96\right)\cdot 101^{6} + \left(61 a + 95\right)\cdot 101^{7} + \left(98 a + 7\right)\cdot 101^{8} + \left(86 a + 47\right)\cdot 101^{9} + \left(63 a + 2\right)\cdot 101^{10} + \left(74 a + 64\right)\cdot 101^{11} + \left(a + 56\right)\cdot 101^{12} + \left(68 a + 58\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 80 a + 87 + \left(88 a + 64\right)\cdot 101 + \left(95 a + 73\right)\cdot 101^{2} + \left(38 a + 46\right)\cdot 101^{3} + \left(7 a + 56\right)\cdot 101^{4} + \left(52 a + 66\right)\cdot 101^{5} + \left(86 a + 48\right)\cdot 101^{6} + \left(20 a + 74\right)\cdot 101^{7} + \left(39 a + 39\right)\cdot 101^{8} + \left(99 a + 73\right)\cdot 101^{9} + \left(60 a + 8\right)\cdot 101^{10} + \left(39 a + 91\right)\cdot 101^{11} + \left(5 a + 4\right)\cdot 101^{12} + \left(77 a + 10\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 a + 13 + \left(56 a + 19\right)\cdot 101 + \left(69 a + 45\right)\cdot 101^{2} + \left(61 a + 19\right)\cdot 101^{3} + \left(2 a + 31\right)\cdot 101^{4} + \left(12 a + 27\right)\cdot 101^{5} + \left(60 a + 70\right)\cdot 101^{6} + \left(39 a + 98\right)\cdot 101^{7} + \left(2 a + 37\right)\cdot 101^{8} + \left(14 a + 94\right)\cdot 101^{9} + \left(37 a + 69\right)\cdot 101^{10} + \left(26 a + 96\right)\cdot 101^{11} + \left(99 a + 89\right)\cdot 101^{12} + \left(32 a + 25\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 84 + \left(7 a + 47\right)\cdot 101 + \left(81 a + 41\right)\cdot 101^{2} + \left(51 a + 2\right)\cdot 101^{3} + \left(96 a + 24\right)\cdot 101^{4} + \left(98 a + 33\right)\cdot 101^{5} + \left(47 a + 25\right)\cdot 101^{6} + \left(80 a + 93\right)\cdot 101^{7} + \left(29 a + 40\right)\cdot 101^{8} + \left(44 a + 28\right)\cdot 101^{9} + \left(68 a + 64\right)\cdot 101^{10} + \left(65 a + 62\right)\cdot 101^{11} + \left(24 a + 86\right)\cdot 101^{12} + \left(82 a + 38\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 3 + \left(12 a + 37\right)\cdot 101 + \left(5 a + 65\right)\cdot 101^{2} + \left(62 a + 5\right)\cdot 101^{3} + \left(93 a + 47\right)\cdot 101^{4} + \left(48 a + 65\right)\cdot 101^{5} + \left(14 a + 39\right)\cdot 101^{6} + \left(80 a + 71\right)\cdot 101^{7} + \left(61 a + 74\right)\cdot 101^{8} + \left(a + 27\right)\cdot 101^{9} + \left(40 a + 52\right)\cdot 101^{10} + \left(61 a + 87\right)\cdot 101^{11} + \left(95 a + 87\right)\cdot 101^{12} + \left(23 a + 9\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a + 2 + \left(93 a + 47\right)\cdot 101 + \left(19 a + 55\right)\cdot 101^{2} + \left(49 a + 27\right)\cdot 101^{3} + \left(4 a + 55\right)\cdot 101^{4} + \left(2 a + 29\right)\cdot 101^{5} + \left(53 a + 17\right)\cdot 101^{6} + \left(20 a + 64\right)\cdot 101^{7} + \left(71 a + 79\right)\cdot 101^{8} + \left(56 a + 74\right)\cdot 101^{9} + \left(32 a + 91\right)\cdot 101^{10} + \left(35 a + 54\right)\cdot 101^{11} + \left(76 a + 18\right)\cdot 101^{12} + \left(18 a + 40\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 a + 9 + \left(83 a + 67\right)\cdot 101 + \left(39 a + 41\right)\cdot 101^{2} + \left(63 a + 97\right)\cdot 101^{3} + \left(93 a + 100\right)\cdot 101^{4} + \left(41 a + 67\right)\cdot 101^{5} + \left(48 a + 27\right)\cdot 101^{6} + \left(20 a + 37\right)\cdot 101^{7} + \left(a + 69\right)\cdot 101^{8} + \left(57 a + 16\right)\cdot 101^{9} + \left(20 a + 95\right)\cdot 101^{10} + \left(4 a + 25\right)\cdot 101^{11} + \left(34 a + 14\right)\cdot 101^{12} + \left(82 a + 13\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 a + 47 + \left(17 a + 38\right)\cdot 101 + \left(61 a + 16\right)\cdot 101^{2} + \left(37 a + 8\right)\cdot 101^{3} + \left(7 a + 8\right)\cdot 101^{4} + \left(59 a + 41\right)\cdot 101^{5} + \left(52 a + 78\right)\cdot 101^{6} + \left(80 a + 70\right)\cdot 101^{7} + \left(99 a + 53\right)\cdot 101^{8} + \left(43 a + 41\right)\cdot 101^{9} + \left(80 a + 19\right)\cdot 101^{10} + \left(96 a + 22\right)\cdot 101^{11} + \left(66 a + 45\right)\cdot 101^{12} + \left(18 a + 5\right)\cdot 101^{13} +O\left(101^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(4,7)$ |
| $(1,7,5,4)(2,8,3,6)$ |
| $(2,3)(4,7)$ |
| $(1,5)(6,8)$ |
| $(1,5)(2,7,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-4$ |
| $2$ | $2$ | $(1,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,3)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,6)(5,8)$ | $2$ |
| $4$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $4$ | $4$ | $(1,7,5,4)(2,8,3,6)$ | $0$ |
| $4$ | $4$ | $(1,4,5,7)(2,8,3,6)$ | $0$ |
| $4$ | $4$ | $(1,6,5,8)(2,4,3,7)$ | $0$ |
| $8$ | $4$ | $(1,3,6,7)(2,8,4,5)$ | $0$ |
| $8$ | $4$ | $(1,6,5,8)(2,3)$ | $0$ |
| $8$ | $4$ | $(1,2,8,4)(3,6,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
