Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 16.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{3} + 9 x + 76 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 51 a^{2} + 26 a + 69 + \left(63 a^{2} + 41 a + 65\right)\cdot 79 + \left(a^{2} + 71 a + 10\right)\cdot 79^{2} + \left(21 a^{2} + 63 a + 47\right)\cdot 79^{3} + \left(15 a^{2} + 37 a + 12\right)\cdot 79^{4} + \left(60 a^{2} + 26 a + 45\right)\cdot 79^{5} + \left(18 a^{2} + 66 a + 33\right)\cdot 79^{6} + \left(61 a^{2} + 3 a + 51\right)\cdot 79^{7} + \left(51 a^{2} + 13 a + 73\right)\cdot 79^{8} + \left(71 a^{2} + 64 a + 34\right)\cdot 79^{9} + \left(2 a^{2} + 70 a + 17\right)\cdot 79^{10} + \left(16 a^{2} + 10 a + 17\right)\cdot 79^{11} + \left(12 a^{2} + 20 a + 73\right)\cdot 79^{12} + \left(33 a^{2} + 75 a + 40\right)\cdot 79^{13} + \left(70 a^{2} + 73 a + 27\right)\cdot 79^{14} + \left(50 a^{2} + 7 a + 68\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 a^{2} + 10 a + 75 + \left(76 a^{2} + 24 a + 64\right)\cdot 79 + \left(59 a^{2} + 30 a + 43\right)\cdot 79^{2} + \left(44 a^{2} + 12 a + 31\right)\cdot 79^{3} + \left(37 a^{2} + 41 a + 67\right)\cdot 79^{4} + \left(38 a^{2} + 35 a + 72\right)\cdot 79^{5} + \left(28 a^{2} + 74 a + 12\right)\cdot 79^{6} + \left(44 a^{2} + 35 a + 29\right)\cdot 79^{7} + \left(56 a^{2} + 8 a + 23\right)\cdot 79^{8} + \left(62 a^{2} + 13 a + 60\right)\cdot 79^{9} + \left(16 a^{2} + 12 a + 21\right)\cdot 79^{10} + \left(6 a^{2} + 4 a + 37\right)\cdot 79^{11} + \left(59 a^{2} + 54 a + 38\right)\cdot 79^{12} + \left(51 a^{2} + 61 a + 73\right)\cdot 79^{13} + \left(31 a^{2} + 60 a + 31\right)\cdot 79^{14} + \left(58 a^{2} + 7 a + 34\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 27\cdot 79 + 53\cdot 79^{2} + 55\cdot 79^{3} + 32\cdot 79^{4} + 10\cdot 79^{5} + 51\cdot 79^{6} + 3\cdot 79^{7} + 56\cdot 79^{8} + 36\cdot 79^{9} + 59\cdot 79^{10} + 7\cdot 79^{11} + 72\cdot 79^{12} + 38\cdot 79^{13} + 71\cdot 79^{14} + 32\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 a^{2} + 45 a + 7 + \left(25 a^{2} + 62 a + 76\right)\cdot 79 + \left(36 a^{2} + 66 a + 59\right)\cdot 79^{2} + \left(66 a^{2} + 40 a + 3\right)\cdot 79^{3} + \left(71 a^{2} + 23 a + 36\right)\cdot 79^{4} + \left(23 a^{2} + 60 a + 64\right)\cdot 79^{5} + \left(29 a^{2} + 51 a + 17\right)\cdot 79^{6} + \left(36 a^{2} + 17 a + 60\right)\cdot 79^{7} + \left(71 a^{2} + 46 a + 33\right)\cdot 79^{8} + \left(22 a^{2} + 77 a + 58\right)\cdot 79^{9} + \left(a^{2} + 8 a + 7\right)\cdot 79^{10} + \left(19 a^{2} + 41 a + 35\right)\cdot 79^{11} + \left(4 a^{2} + 42 a + 25\right)\cdot 79^{12} + \left(76 a^{2} + 34 a + 61\right)\cdot 79^{13} + \left(44 a^{2} + 32 a + 32\right)\cdot 79^{14} + \left(55 a^{2} + 52 a + 17\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a^{2} + 72 a + 66 + \left(66 a^{2} + 18 a + 1\right)\cdot 79 + \left(13 a^{2} + 12 a + 4\right)\cdot 79^{2} + \left(44 a^{2} + 22 a + 28\right)\cdot 79^{3} + \left(46 a^{2} + 28 a + 42\right)\cdot 79^{4} + \left(13 a^{2} + 42 a + 2\right)\cdot 79^{5} + \left(55 a^{2} + 34 a + 15\right)\cdot 79^{6} + \left(18 a^{2} + 46 a + 33\right)\cdot 79^{7} + \left(7 a^{2} + 63 a + 43\right)\cdot 79^{8} + \left(12 a^{2} + 8 a + 72\right)\cdot 79^{9} + \left(23 a^{2} + 70 a + 59\right)\cdot 79^{10} + \left(25 a^{2} + 76 a + 72\right)\cdot 79^{11} + \left(53 a^{2} + 62 a + 3\right)\cdot 79^{12} + \left(53 a^{2} + 22 a + 6\right)\cdot 79^{13} + \left(5 a^{2} + 34 a + 34\right)\cdot 79^{14} + \left(2 a^{2} + 30 a + 12\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 54\cdot 79 + 68\cdot 79^{2} + 50\cdot 79^{3} + 6\cdot 79^{4} + 35\cdot 79^{5} + 70\cdot 79^{6} + 72\cdot 79^{7} + 72\cdot 79^{8} + 34\cdot 79^{9} + 46\cdot 79^{10} + 34\cdot 79^{11} + 39\cdot 79^{12} + 74\cdot 79^{13} + 36\cdot 79^{14} + 45\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 a^{2} + 8 a + 3 + \left(68 a^{2} + 54 a + 16\right)\cdot 79 + \left(40 a^{2} + 19 a + 8\right)\cdot 79^{2} + \left(70 a^{2} + 53 a + 28\right)\cdot 79^{3} + \left(70 a^{2} + 17 a + 30\right)\cdot 79^{4} + \left(73 a^{2} + 71 a + 48\right)\cdot 79^{5} + \left(30 a^{2} + 39 a + 27\right)\cdot 79^{6} + \left(60 a^{2} + 57 a + 46\right)\cdot 79^{7} + \left(34 a^{2} + 19 a + 50\right)\cdot 79^{8} + \left(63 a^{2} + 16 a + 64\right)\cdot 79^{9} + \left(74 a^{2} + 78 a + 53\right)\cdot 79^{10} + \left(43 a^{2} + 26 a + 26\right)\cdot 79^{11} + \left(62 a^{2} + 16 a + 59\right)\cdot 79^{12} + \left(48 a^{2} + 48 a + 55\right)\cdot 79^{13} + \left(42 a^{2} + 51 a + 18\right)\cdot 79^{14} + \left(51 a^{2} + 18 a + 72\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 a^{2} + 76 a + 17 + \left(15 a^{2} + 35 a + 12\right)\cdot 79 + \left(5 a^{2} + 36 a + 31\right)\cdot 79^{2} + \left(69 a^{2} + 44 a + 19\right)\cdot 79^{3} + \left(73 a^{2} + 9 a + 48\right)\cdot 79^{4} + \left(26 a^{2} + a + 3\right)\cdot 79^{5} + \left(74 a^{2} + 49 a + 51\right)\cdot 79^{6} + \left(15 a^{2} + 75 a + 16\right)\cdot 79^{7} + \left(15 a^{2} + 6 a + 12\right)\cdot 79^{8} + \left(4 a^{2} + 57 a + 25\right)\cdot 79^{9} + \left(39 a^{2} + 75 a + 76\right)\cdot 79^{10} + \left(47 a^{2} + 76 a + 47\right)\cdot 79^{11} + \left(45 a^{2} + 40 a + 36\right)\cdot 79^{12} + \left(52 a^{2} + 73 a + 78\right)\cdot 79^{13} + \left(41 a^{2} + 62 a + 12\right)\cdot 79^{14} + \left(18 a^{2} + 40 a + 32\right)\cdot 79^{15} +O\left(79^{ 16 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 60 + 75\cdot 79 + 35\cdot 79^{2} + 51\cdot 79^{3} + 39\cdot 79^{4} + 33\cdot 79^{5} + 36\cdot 79^{6} + 2\cdot 79^{7} + 29\cdot 79^{8} + 7\cdot 79^{9} + 52\cdot 79^{10} + 36\cdot 79^{11} + 46\cdot 79^{12} + 44\cdot 79^{13} + 49\cdot 79^{14} +O\left(79^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,8)(2,4,6)(5,7,9)$ |
| $(1,4,7)(2,5,8)(3,6,9)$ |
| $(2,3)(4,7)(5,9)(6,8)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $-3$ |
| $3$ | $3$ | $(1,7,4)(2,5,8)$ | $0$ |
| $3$ | $3$ | $(1,4,7)(2,8,5)$ | $0$ |
| $6$ | $3$ | $(1,3,8)(2,4,6)(5,7,9)$ | $0$ |
| $6$ | $3$ | $(1,8,3)(2,6,4)(5,9,7)$ | $0$ |
| $6$ | $3$ | $(1,8,6)(2,9,4)(3,7,5)$ | $0$ |
| $9$ | $6$ | $(1,2,7,5,4,8)(3,6)$ | $0$ |
| $9$ | $6$ | $(1,8,4,5,7,2)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
