Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 23.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 73 a + 95 + \left(13 a + 72\right)\cdot 97 + \left(93 a + 76\right)\cdot 97^{2} + \left(94 a + 96\right)\cdot 97^{3} + \left(49 a + 77\right)\cdot 97^{4} + \left(60 a + 48\right)\cdot 97^{5} + \left(26 a + 73\right)\cdot 97^{6} + \left(55 a + 5\right)\cdot 97^{7} + \left(68 a + 28\right)\cdot 97^{8} + \left(96 a + 27\right)\cdot 97^{9} + \left(32 a + 34\right)\cdot 97^{10} + \left(93 a + 44\right)\cdot 97^{11} + \left(44 a + 52\right)\cdot 97^{12} + \left(22 a + 54\right)\cdot 97^{13} + \left(91 a + 41\right)\cdot 97^{14} + \left(70 a + 94\right)\cdot 97^{15} + \left(22 a + 10\right)\cdot 97^{16} + \left(79 a + 57\right)\cdot 97^{17} + \left(37 a + 25\right)\cdot 97^{18} + \left(28 a + 47\right)\cdot 97^{19} + \left(93 a + 20\right)\cdot 97^{20} + \left(54 a + 15\right)\cdot 97^{21} + \left(36 a + 5\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 85\cdot 97 + 45\cdot 97^{2} + 18\cdot 97^{3} + 39\cdot 97^{4} + 58\cdot 97^{5} + 51\cdot 97^{6} + 94\cdot 97^{7} + 61\cdot 97^{8} + 2\cdot 97^{9} + 52\cdot 97^{10} + 93\cdot 97^{11} + 27\cdot 97^{12} + 53\cdot 97^{13} + 79\cdot 97^{15} + 55\cdot 97^{16} + 54\cdot 97^{17} + 16\cdot 97^{18} + 46\cdot 97^{19} + 4\cdot 97^{20} + 68\cdot 97^{21} + 20\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 77 + 11\cdot 97 + 51\cdot 97^{2} + 78\cdot 97^{3} + 57\cdot 97^{4} + 38\cdot 97^{5} + 45\cdot 97^{6} + 2\cdot 97^{7} + 35\cdot 97^{8} + 94\cdot 97^{9} + 44\cdot 97^{10} + 3\cdot 97^{11} + 69\cdot 97^{12} + 43\cdot 97^{13} + 96\cdot 97^{14} + 17\cdot 97^{15} + 41\cdot 97^{16} + 42\cdot 97^{17} + 80\cdot 97^{18} + 50\cdot 97^{19} + 92\cdot 97^{20} + 28\cdot 97^{21} + 76\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 71 + \left(83 a + 13\right)\cdot 97 + \left(3 a + 59\right)\cdot 97^{2} + \left(2 a + 1\right)\cdot 97^{3} + \left(47 a + 33\right)\cdot 97^{4} + \left(36 a + 59\right)\cdot 97^{5} + \left(70 a + 39\right)\cdot 97^{6} + \left(41 a + 34\right)\cdot 97^{7} + \left(28 a + 41\right)\cdot 97^{8} + 55\cdot 97^{9} + \left(64 a + 67\right)\cdot 97^{10} + \left(3 a + 7\right)\cdot 97^{11} + \left(52 a + 4\right)\cdot 97^{12} + \left(74 a + 32\right)\cdot 97^{13} + \left(5 a + 13\right)\cdot 97^{14} + \left(26 a + 74\right)\cdot 97^{15} + \left(74 a + 59\right)\cdot 97^{16} + \left(17 a + 16\right)\cdot 97^{17} + \left(59 a + 81\right)\cdot 97^{18} + \left(68 a + 37\right)\cdot 97^{19} + \left(3 a + 85\right)\cdot 97^{20} + \left(42 a + 73\right)\cdot 97^{21} + \left(60 a + 83\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 73 a + 27 + \left(13 a + 83\right)\cdot 97 + \left(93 a + 37\right)\cdot 97^{2} + \left(94 a + 95\right)\cdot 97^{3} + \left(49 a + 63\right)\cdot 97^{4} + \left(60 a + 37\right)\cdot 97^{5} + \left(26 a + 57\right)\cdot 97^{6} + \left(55 a + 62\right)\cdot 97^{7} + \left(68 a + 55\right)\cdot 97^{8} + \left(96 a + 41\right)\cdot 97^{9} + \left(32 a + 29\right)\cdot 97^{10} + \left(93 a + 89\right)\cdot 97^{11} + \left(44 a + 92\right)\cdot 97^{12} + \left(22 a + 64\right)\cdot 97^{13} + \left(91 a + 83\right)\cdot 97^{14} + \left(70 a + 22\right)\cdot 97^{15} + \left(22 a + 37\right)\cdot 97^{16} + \left(79 a + 80\right)\cdot 97^{17} + \left(37 a + 15\right)\cdot 97^{18} + \left(28 a + 59\right)\cdot 97^{19} + \left(93 a + 11\right)\cdot 97^{20} + \left(54 a + 23\right)\cdot 97^{21} + \left(36 a + 13\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 3 + \left(83 a + 24\right)\cdot 97 + \left(3 a + 20\right)\cdot 97^{2} + 2 a\cdot 97^{3} + \left(47 a + 19\right)\cdot 97^{4} + \left(36 a + 48\right)\cdot 97^{5} + \left(70 a + 23\right)\cdot 97^{6} + \left(41 a + 91\right)\cdot 97^{7} + \left(28 a + 68\right)\cdot 97^{8} + 69\cdot 97^{9} + \left(64 a + 62\right)\cdot 97^{10} + \left(3 a + 52\right)\cdot 97^{11} + \left(52 a + 44\right)\cdot 97^{12} + \left(74 a + 42\right)\cdot 97^{13} + \left(5 a + 55\right)\cdot 97^{14} + \left(26 a + 2\right)\cdot 97^{15} + \left(74 a + 86\right)\cdot 97^{16} + \left(17 a + 39\right)\cdot 97^{17} + \left(59 a + 71\right)\cdot 97^{18} + \left(68 a + 49\right)\cdot 97^{19} + \left(3 a + 76\right)\cdot 97^{20} + \left(42 a + 81\right)\cdot 97^{21} + \left(60 a + 91\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 a + 57 + \left(87 a + 93\right)\cdot 97 + \left(39 a + 23\right)\cdot 97^{2} + \left(88 a + 24\right)\cdot 97^{3} + \left(47 a + 20\right)\cdot 97^{4} + 47 a\cdot 97^{5} + \left(56 a + 44\right)\cdot 97^{6} + \left(82 a + 35\right)\cdot 97^{7} + \left(94 a + 42\right)\cdot 97^{8} + \left(32 a + 79\right)\cdot 97^{9} + \left(42 a + 43\right)\cdot 97^{10} + \left(60 a + 39\right)\cdot 97^{11} + \left(16 a + 70\right)\cdot 97^{12} + \left(69 a + 70\right)\cdot 97^{13} + \left(27 a + 20\right)\cdot 97^{14} + \left(29 a + 96\right)\cdot 97^{15} + \left(37 a + 92\right)\cdot 97^{16} + \left(28 a + 52\right)\cdot 97^{17} + \left(84 a + 20\right)\cdot 97^{18} + \left(86 a + 47\right)\cdot 97^{19} + \left(57 a + 14\right)\cdot 97^{20} + \left(21 a + 18\right)\cdot 97^{21} + \left(70 a + 24\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 a + 41 + \left(9 a + 3\right)\cdot 97 + \left(57 a + 73\right)\cdot 97^{2} + \left(8 a + 72\right)\cdot 97^{3} + \left(49 a + 76\right)\cdot 97^{4} + \left(49 a + 96\right)\cdot 97^{5} + \left(40 a + 52\right)\cdot 97^{6} + \left(14 a + 61\right)\cdot 97^{7} + \left(2 a + 54\right)\cdot 97^{8} + \left(64 a + 17\right)\cdot 97^{9} + \left(54 a + 53\right)\cdot 97^{10} + \left(36 a + 57\right)\cdot 97^{11} + \left(80 a + 26\right)\cdot 97^{12} + \left(27 a + 26\right)\cdot 97^{13} + \left(69 a + 76\right)\cdot 97^{14} + 67 a\cdot 97^{15} + \left(59 a + 4\right)\cdot 97^{16} + \left(68 a + 44\right)\cdot 97^{17} + \left(12 a + 76\right)\cdot 97^{18} + \left(10 a + 49\right)\cdot 97^{19} + \left(39 a + 82\right)\cdot 97^{20} + \left(75 a + 78\right)\cdot 97^{21} + \left(26 a + 72\right)\cdot 97^{22} +O\left(97^{ 23 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,7,5,6,8,4)(2,3)$ |
| $(1,2,6,3)(4,8,5,7)$ |
| $(2,5,7)(3,4,8)$ |
| $(1,8,4,3,6,7,5,2)$ |
| $(1,6)(2,3)(4,5)(7,8)$ |
| $(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-8$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $0$ |
| $12$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $24$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $32$ | $3$ | $(1,5,8)(4,7,6)$ | $-1$ |
| $6$ | $4$ | $(1,2,6,3)(4,8,5,7)$ | $0$ |
| $6$ | $4$ | $(1,4,6,5)(2,8,3,7)$ | $0$ |
| $12$ | $4$ | $(1,6)(2,4,3,5)(7,8)$ | $0$ |
| $12$ | $4$ | $(2,5,3,4)$ | $0$ |
| $32$ | $6$ | $(1,7,5,6,8,4)(2,3)$ | $1$ |
| $24$ | $8$ | $(1,8,4,3,6,7,5,2)$ | $0$ |
| $24$ | $8$ | $(1,5,2,7,6,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
