Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 21.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 44 + \left(39 a + 5\right)\cdot 79 + \left(47 a + 68\right)\cdot 79^{2} + \left(27 a + 42\right)\cdot 79^{3} + \left(47 a + 59\right)\cdot 79^{4} + \left(61 a + 6\right)\cdot 79^{5} + \left(31 a + 18\right)\cdot 79^{6} + \left(72 a + 49\right)\cdot 79^{7} + \left(57 a + 76\right)\cdot 79^{8} + \left(36 a + 67\right)\cdot 79^{9} + \left(60 a + 22\right)\cdot 79^{10} + \left(29 a + 51\right)\cdot 79^{11} + \left(23 a + 58\right)\cdot 79^{12} + \left(a + 65\right)\cdot 79^{13} + \left(55 a + 19\right)\cdot 79^{14} + \left(8 a + 56\right)\cdot 79^{15} + \left(22 a + 25\right)\cdot 79^{16} + \left(2 a + 10\right)\cdot 79^{17} + \left(33 a + 50\right)\cdot 79^{18} + \left(48 a + 52\right)\cdot 79^{19} + \left(72 a + 1\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 a + 72 + \left(15 a + 11\right)\cdot 79 + \left(53 a + 40\right)\cdot 79^{2} + \left(25 a + 39\right)\cdot 79^{3} + \left(57 a + 25\right)\cdot 79^{4} + \left(73 a + 11\right)\cdot 79^{5} + \left(62 a + 51\right)\cdot 79^{6} + \left(24 a + 1\right)\cdot 79^{7} + \left(45 a + 35\right)\cdot 79^{8} + \left(76 a + 71\right)\cdot 79^{9} + \left(17 a + 46\right)\cdot 79^{10} + \left(63 a + 12\right)\cdot 79^{11} + \left(34 a + 3\right)\cdot 79^{12} + \left(73 a + 65\right)\cdot 79^{13} + \left(51 a + 78\right)\cdot 79^{14} + \left(3 a + 59\right)\cdot 79^{15} + \left(50 a + 35\right)\cdot 79^{16} + \left(47 a + 16\right)\cdot 79^{17} + \left(16 a + 38\right)\cdot 79^{18} + \left(40 a + 30\right)\cdot 79^{19} + \left(6 a + 11\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 6\cdot 79 + 46\cdot 79^{2} + 26\cdot 79^{3} + 55\cdot 79^{4} + 62\cdot 79^{5} + 35\cdot 79^{6} + 36\cdot 79^{7} + 19\cdot 79^{8} + 68\cdot 79^{9} + 56\cdot 79^{10} + 43\cdot 79^{11} + 75\cdot 79^{12} + 46\cdot 79^{13} + 46\cdot 79^{14} + 20\cdot 79^{15} + 76\cdot 79^{16} + 57\cdot 79^{17} + 70\cdot 79^{18} + 53\cdot 79^{19} + 15\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 19 + \left(39 a + 70\right)\cdot 79 + \left(31 a + 75\right)\cdot 79^{2} + \left(51 a + 22\right)\cdot 79^{3} + 31 a\cdot 79^{4} + \left(17 a + 21\right)\cdot 79^{5} + \left(47 a + 67\right)\cdot 79^{6} + \left(6 a + 10\right)\cdot 79^{7} + \left(21 a + 62\right)\cdot 79^{8} + \left(42 a + 46\right)\cdot 79^{9} + \left(18 a + 46\right)\cdot 79^{10} + \left(49 a + 20\right)\cdot 79^{11} + \left(55 a + 52\right)\cdot 79^{12} + \left(77 a + 43\right)\cdot 79^{13} + \left(23 a + 73\right)\cdot 79^{14} + \left(70 a + 9\right)\cdot 79^{15} + \left(56 a + 39\right)\cdot 79^{16} + \left(76 a + 69\right)\cdot 79^{17} + \left(45 a + 1\right)\cdot 79^{18} + \left(30 a + 68\right)\cdot 79^{19} + \left(6 a + 25\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 a + 46 + \left(28 a + 38\right)\cdot 79 + \left(40 a + 42\right)\cdot 79^{2} + \left(74 a + 27\right)\cdot 79^{3} + \left(32 a + 48\right)\cdot 79^{4} + \left(48 a + 33\right)\cdot 79^{5} + \left(52 a + 56\right)\cdot 79^{6} + \left(53 a + 19\right)\cdot 79^{7} + \left(21 a + 57\right)\cdot 79^{8} + \left(56 a + 69\right)\cdot 79^{9} + \left(a + 76\right)\cdot 79^{10} + \left(13 a + 14\right)\cdot 79^{11} + \left(50 a + 28\right)\cdot 79^{12} + \left(59 a + 41\right)\cdot 79^{13} + \left(70 a + 70\right)\cdot 79^{14} + 26 a\cdot 79^{15} + \left(58 a + 8\right)\cdot 79^{16} + \left(19 a + 39\right)\cdot 79^{17} + \left(29 a + 24\right)\cdot 79^{18} + \left(5 a + 28\right)\cdot 79^{19} + \left(65 a + 55\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 67\cdot 79 + 69\cdot 79^{2} + 3\cdot 79^{3} + 63\cdot 79^{4} + 24\cdot 79^{5} + 65\cdot 79^{6} + 55\cdot 79^{7} + 63\cdot 79^{8} + 21\cdot 79^{9} + 55\cdot 79^{10} + 9\cdot 79^{11} + 58\cdot 79^{12} + 56\cdot 79^{13} + 45\cdot 79^{14} + 41\cdot 79^{15} + 9\cdot 79^{16} + 29\cdot 79^{17} + 10\cdot 79^{18} + 24\cdot 79^{19} + 34\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 a + 53 + \left(63 a + 46\right)\cdot 79 + \left(25 a + 77\right)\cdot 79^{2} + \left(53 a + 11\right)\cdot 79^{3} + \left(21 a + 57\right)\cdot 79^{4} + \left(5 a + 27\right)\cdot 79^{5} + \left(16 a + 40\right)\cdot 79^{6} + \left(54 a + 42\right)\cdot 79^{7} + \left(33 a + 55\right)\cdot 79^{8} + \left(2 a + 23\right)\cdot 79^{9} + \left(61 a + 67\right)\cdot 79^{10} + \left(15 a + 57\right)\cdot 79^{11} + \left(44 a + 53\right)\cdot 79^{12} + \left(5 a + 24\right)\cdot 79^{13} + \left(27 a + 57\right)\cdot 79^{14} + \left(75 a + 11\right)\cdot 79^{15} + \left(28 a + 3\right)\cdot 79^{16} + \left(31 a + 14\right)\cdot 79^{17} + \left(62 a + 7\right)\cdot 79^{18} + \left(38 a + 54\right)\cdot 79^{19} + \left(72 a + 56\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ a + 45 + \left(50 a + 68\right)\cdot 79 + \left(38 a + 53\right)\cdot 79^{2} + \left(4 a + 61\right)\cdot 79^{3} + \left(46 a + 6\right)\cdot 79^{4} + \left(30 a + 49\right)\cdot 79^{5} + \left(26 a + 60\right)\cdot 79^{6} + \left(25 a + 20\right)\cdot 79^{7} + \left(57 a + 25\right)\cdot 79^{8} + \left(22 a + 25\right)\cdot 79^{9} + \left(77 a + 22\right)\cdot 79^{10} + \left(65 a + 26\right)\cdot 79^{11} + \left(28 a + 65\right)\cdot 79^{12} + \left(19 a + 50\right)\cdot 79^{13} + \left(8 a + 2\right)\cdot 79^{14} + \left(52 a + 36\right)\cdot 79^{15} + \left(20 a + 39\right)\cdot 79^{16} + 59 a\cdot 79^{17} + \left(49 a + 34\right)\cdot 79^{18} + \left(73 a + 4\right)\cdot 79^{19} + \left(13 a + 36\right)\cdot 79^{20} +O\left(79^{ 21 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,5)$ |
| $(1,4,3,2)(5,6,7,8)$ |
| $(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $4$ | $2$ | $(1,8)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)(4,5)$ | $-2$ |
| $6$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(3,4)(5,6)$ | $2$ |
| $12$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-2$ |
| $24$ | $2$ | $(1,8)(3,4)(5,6)$ | $0$ |
| $32$ | $3$ | $(1,4,2)(5,7,8)$ | $1$ |
| $12$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $12$ | $4$ | $(3,5,6,4)$ | $2$ |
| $12$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $-2$ |
| $24$ | $4$ | $(1,6,8,3)(2,4)(5,7)$ | $0$ |
| $24$ | $4$ | $(1,8)(3,5,6,4)$ | $0$ |
| $48$ | $4$ | $(1,4,3,2)(5,6,7,8)$ | $0$ |
| $32$ | $6$ | $(1,5,7,8,4,2)$ | $1$ |
| $32$ | $6$ | $(1,4,2)(3,6)(5,7,8)$ | $-1$ |
| $32$ | $6$ | $(1,5,7,8,4,2)(3,6)$ | $-1$ |
| $48$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
