Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 a^{2} + 37 a + 61 + \left(21 a^{2} + 46 a + 18\right)\cdot 67 + \left(47 a^{2} + 58 a + 55\right)\cdot 67^{2} + \left(25 a^{2} + 8 a + 35\right)\cdot 67^{3} + \left(33 a^{2} + 39 a + 66\right)\cdot 67^{4} + \left(35 a^{2} + 55 a + 7\right)\cdot 67^{5} + \left(18 a^{2} + 13 a + 7\right)\cdot 67^{6} + \left(53 a^{2} + 28 a + 12\right)\cdot 67^{7} + \left(16 a^{2} + 51 a\right)\cdot 67^{8} + \left(21 a^{2} + a + 18\right)\cdot 67^{9} + \left(46 a^{2} + 31 a + 51\right)\cdot 67^{10} + \left(51 a^{2} + 16 a + 5\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a^{2} + 9 a + 64 + \left(19 a^{2} + 3 a + 9\right)\cdot 67 + \left(52 a^{2} + 23 a + 8\right)\cdot 67^{2} + \left(45 a^{2} + 54 a + 49\right)\cdot 67^{3} + \left(18 a^{2} + 38 a + 7\right)\cdot 67^{4} + \left(11 a^{2} + 57 a + 45\right)\cdot 67^{5} + \left(54 a^{2} + 43 a + 15\right)\cdot 67^{6} + \left(45 a^{2} + 4 a + 49\right)\cdot 67^{7} + \left(39 a^{2} + 5 a + 24\right)\cdot 67^{8} + \left(3 a^{2} + 6 a + 14\right)\cdot 67^{9} + \left(47 a^{2} + 55 a + 54\right)\cdot 67^{10} + \left(26 a^{2} + 28 a + 39\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 4\cdot 67 + 11\cdot 67^{2} + 38\cdot 67^{3} + 24\cdot 67^{4} + 6\cdot 67^{5} + 14\cdot 67^{6} + 34\cdot 67^{7} + 50\cdot 67^{8} + 58\cdot 67^{9} + 15\cdot 67^{10} + 54\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a^{2} + 34 a + 26 + \left(10 a^{2} + 17 a + 42\right)\cdot 67 + \left(8 a^{2} + 35 a + 32\right)\cdot 67^{2} + \left(29 a^{2} + 56 a + 49\right)\cdot 67^{3} + \left(38 a^{2} + 2 a + 19\right)\cdot 67^{4} + \left(31 a^{2} + 57 a + 59\right)\cdot 67^{5} + \left(40 a^{2} + 58 a + 27\right)\cdot 67^{6} + \left(3 a^{2} + 40 a + 14\right)\cdot 67^{7} + \left(48 a^{2} + 17 a + 58\right)\cdot 67^{8} + \left(18 a^{2} + 21 a + 7\right)\cdot 67^{9} + \left(26 a^{2} + 30 a + 38\right)\cdot 67^{10} + \left(24 a^{2} + 17 a + 30\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 34\cdot 67 + 64\cdot 67^{2} + 63\cdot 67^{3} + 28\cdot 67^{4} + 36\cdot 67^{5} + 18\cdot 67^{6} + 36\cdot 67^{7} + 42\cdot 67^{8} + 62\cdot 67^{9} + 56\cdot 67^{10} + 45\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a^{2} + 21 a + 2 + \left(17 a^{2} + 37 a + 3\right)\cdot 67 + \left(56 a^{2} + 51 a + 24\right)\cdot 67^{2} + \left(47 a^{2} + 2 a + 57\right)\cdot 67^{3} + \left(58 a^{2} + 42 a + 33\right)\cdot 67^{4} + \left(44 a^{2} + 39 a + 45\right)\cdot 67^{5} + \left(8 a^{2} + 10 a + 34\right)\cdot 67^{6} + \left(29 a^{2} + 48 a + 49\right)\cdot 67^{7} + \left(59 a^{2} + 16 a + 36\right)\cdot 67^{8} + \left(28 a^{2} + 46 a + 48\right)\cdot 67^{9} + \left(35 a^{2} + 24 a + 7\right)\cdot 67^{10} + \left(15 a^{2} + 62\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 + 28\cdot 67 + 58\cdot 67^{2} + 31\cdot 67^{3} + 13\cdot 67^{4} + 24\cdot 67^{5} + 34\cdot 67^{6} + 63\cdot 67^{7} + 40\cdot 67^{8} + 12\cdot 67^{9} + 61\cdot 67^{10} + 33\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 37 a + 1 + \left(30 a^{2} + 26 a + 54\right)\cdot 67 + \left(25 a^{2} + 59 a + 34\right)\cdot 67^{2} + \left(40 a^{2} + 9 a + 27\right)\cdot 67^{3} + \left(56 a^{2} + 53 a + 25\right)\cdot 67^{4} + \left(10 a^{2} + 36 a + 43\right)\cdot 67^{5} + \left(4 a^{2} + 12 a + 16\right)\cdot 67^{6} + \left(59 a^{2} + 14 a + 35\right)\cdot 67^{7} + \left(34 a^{2} + 45 a + 5\right)\cdot 67^{8} + \left(34 a^{2} + 14 a + 4\right)\cdot 67^{9} + \left(51 a^{2} + 54 a + 5\right)\cdot 67^{10} + \left(24 a^{2} + 37 a + 32\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 62 a^{2} + 63 a + 47 + \left(34 a^{2} + 2 a + 5\right)\cdot 67 + \left(11 a^{2} + 40 a + 46\right)\cdot 67^{2} + \left(12 a^{2} + a + 48\right)\cdot 67^{3} + \left(62 a^{2} + 25 a + 47\right)\cdot 67^{4} + \left(66 a^{2} + 21 a + 66\right)\cdot 67^{5} + \left(7 a^{2} + 61 a + 31\right)\cdot 67^{6} + \left(10 a^{2} + 64 a + 40\right)\cdot 67^{7} + \left(2 a^{2} + 64 a + 8\right)\cdot 67^{8} + \left(27 a^{2} + 43 a + 41\right)\cdot 67^{9} + \left(61 a^{2} + 5 a + 44\right)\cdot 67^{10} + \left(57 a^{2} + 33 a + 30\right)\cdot 67^{11} +O\left(67^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6,8)(3,5,7)$ |
| $(1,2,5)(3,4,6)(7,9,8)$ |
| $(2,3)(4,9)(5,6)(7,8)$ |
| $(1,9,4)(2,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,7)(2,6)(3,4)(5,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,4)(2,8,6)(3,5,7)$ |
$-3$ |
| $3$ |
$3$ |
$(1,9,4)(2,6,8)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,9)(2,8,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,2,5)(3,4,6)(7,9,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,2)(3,6,4)(7,8,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,7)(2,5,4)(3,9,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,3,9,7,4,5)(2,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,5,4,7,9,3)(2,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
