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