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:
| $r_{ 1 }$ |
$=$ |
$ 72 a^{2} + 30 a + 31 + \left(74 a + 68\right)\cdot 79 + \left(4 a^{2} + 45 a + 45\right)\cdot 79^{2} + \left(11 a^{2} + 29 a + 15\right)\cdot 79^{3} + \left(31 a^{2} + 62 a + 40\right)\cdot 79^{4} + \left(64 a^{2} + 23 a + 5\right)\cdot 79^{5} + \left(39 a^{2} + 53 a + 7\right)\cdot 79^{6} + \left(73 a^{2} + 2 a + 34\right)\cdot 79^{7} + \left(45 a^{2} + 30 a + 62\right)\cdot 79^{8} + \left(55 a^{2} + 21 a + 18\right)\cdot 79^{9} + \left(6 a^{2} + 29 a + 62\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a^{2} + 30 a + 50 + \left(73 a^{2} + 4 a + 39\right)\cdot 79 + \left(21 a^{2} + 75 a + 33\right)\cdot 79^{2} + \left(69 a^{2} + 2 a + 77\right)\cdot 79^{3} + \left(53 a^{2} + 20 a + 74\right)\cdot 79^{4} + \left(74 a^{2} + 17 a + 8\right)\cdot 79^{5} + \left(23 a^{2} + 37 a + 54\right)\cdot 79^{6} + \left(51 a^{2} + 8 a + 21\right)\cdot 79^{7} + \left(65 a^{2} + 51 a + 26\right)\cdot 79^{8} + \left(24 a^{2} + 55 a + 13\right)\cdot 79^{9} + \left(19 a^{2} + 42 a + 51\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a^{2} + 41 a + 75 + \left(10 a^{2} + 72 a + 55\right)\cdot 79 + \left(54 a^{2} + 23 a + 68\right)\cdot 79^{2} + \left(47 a^{2} + 23 a + 26\right)\cdot 79^{3} + \left(72 a^{2} + 74 a + 29\right)\cdot 79^{4} + \left(18 a^{2} + 19 a + 69\right)\cdot 79^{5} + \left(37 a^{2} + 25 a + 54\right)\cdot 79^{6} + \left(37 a^{2} + 57 a + 17\right)\cdot 79^{7} + \left(76 a^{2} + 42 a + 12\right)\cdot 79^{8} + \left(16 a^{2} + 61 a + 45\right)\cdot 79^{9} + \left(76 a^{2} + 28 a + 76\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a^{2} + 8 a + 6 + \left(74 a^{2} + 2 a + 47\right)\cdot 79 + \left(2 a^{2} + 59 a + 77\right)\cdot 79^{2} + \left(41 a^{2} + 52 a + 65\right)\cdot 79^{3} + \left(31 a^{2} + 63 a + 19\right)\cdot 79^{4} + \left(64 a^{2} + 41 a + 26\right)\cdot 79^{5} + \left(17 a^{2} + 16 a + 17\right)\cdot 79^{6} + \left(69 a^{2} + 13 a + 50\right)\cdot 79^{7} + \left(15 a^{2} + 64 a + 43\right)\cdot 79^{8} + \left(37 a^{2} + 40 a + 8\right)\cdot 79^{9} + \left(62 a^{2} + 7 a + 73\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 73 + 60\cdot 79 + 3\cdot 79^{2} + 75\cdot 79^{3} + 54\cdot 79^{4} + 40\cdot 79^{5} + 8\cdot 79^{6} + 35\cdot 79^{7} + 2\cdot 79^{8} + 7\cdot 79^{9} + 23\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a^{2} + 60 a + 70 + \left(35 a^{2} + 43 a + 38\right)\cdot 79 + \left(31 a^{2} + 3 a + 52\right)\cdot 79^{2} + \left(29 a^{2} + 31 a + 46\right)\cdot 79^{3} + \left(42 a^{2} + 17 a + 28\right)\cdot 79^{4} + \left(34 a^{2} + 28 a + 63\right)\cdot 79^{5} + \left(41 a^{2} + 22 a + 16\right)\cdot 79^{6} + \left(57 a^{2} + 10 a + 17\right)\cdot 79^{7} + \left(26 a^{2} + 72 a + 26\right)\cdot 79^{8} + \left(34 a^{2} + 42 a + 49\right)\cdot 79^{9} + \left(a^{2} + 2 a + 30\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 11 + 59\cdot 79 + 44\cdot 79^{2} + 2\cdot 79^{3} + 40\cdot 79^{4} + 14\cdot 79^{5} + 27\cdot 79^{6} + 68\cdot 79^{7} + 14\cdot 79^{8} + 15\cdot 79^{9} + 76\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 47 a^{2} + 68 a + 39 + \left(42 a^{2} + 39 a + 2\right)\cdot 79 + \left(43 a^{2} + 29 a + 46\right)\cdot 79^{2} + \left(38 a^{2} + 18 a + 22\right)\cdot 79^{3} + \left(5 a^{2} + 78 a + 44\right)\cdot 79^{4} + \left(59 a^{2} + 26 a + 52\right)\cdot 79^{5} + \left(76 a^{2} + 3 a + 70\right)\cdot 79^{6} + \left(26 a^{2} + 66 a + 70\right)\cdot 79^{7} + \left(6 a^{2} + 55 a + 61\right)\cdot 79^{8} + \left(68 a^{2} + 14 a + 14\right)\cdot 79^{9} + \left(70 a^{2} + 47 a + 52\right)\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 41 + 22\cdot 79 + 22\cdot 79^{2} + 62\cdot 79^{3} + 62\cdot 79^{4} + 34\cdot 79^{5} + 59\cdot 79^{6} + 66\cdot 79^{8} + 64\cdot 79^{9} + 28\cdot 79^{10} +O\left(79^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,6)(2,3,4)(5,7,9)$ |
| $(2,4,3)(5,7,9)$ |
| $(2,9)(3,5)(4,7)$ |
| $(2,4)(5,7)(6,8)$ |
| $(1,7,4)(2,8,9)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,8)(3,4)(5,9)$ | $0$ |
| $9$ | $2$ | $(2,9)(3,5)(4,7)$ | $2$ |
| $9$ | $2$ | $(1,9)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,8,6)(2,3,4)(5,7,9)$ | $-3$ |
| $6$ | $3$ | $(1,7,4)(2,8,9)(3,6,5)$ | $0$ |
| $6$ | $3$ | $(1,8,6)(5,9,7)$ | $0$ |
| $12$ | $3$ | $(1,3,7)(2,5,6)(4,9,8)$ | $0$ |
| $18$ | $6$ | $(1,5,3,8,9,4)(2,6,7)$ | $0$ |
| $18$ | $6$ | $(1,8,6)(2,5,4,9,3,7)$ | $-1$ |
| $18$ | $6$ | $(1,5,8,9,6,7)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
