Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 3\cdot 67 + 21\cdot 67^{2} + 26\cdot 67^{3} + 16\cdot 67^{4} + 56\cdot 67^{5} + 48\cdot 67^{6} + 54\cdot 67^{7} + 37\cdot 67^{8} + 48\cdot 67^{9} + 20\cdot 67^{10} + 52\cdot 67^{11} + 16\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 a^{2} + 40 a + 11 + \left(3 a^{2} + 45 a + 15\right)\cdot 67 + \left(26 a^{2} + 61 a + 37\right)\cdot 67^{2} + \left(63 a^{2} + 13 a + 52\right)\cdot 67^{3} + \left(39 a + 3\right)\cdot 67^{4} + \left(27 a^{2} + 41\right)\cdot 67^{5} + \left(18 a^{2} + 35 a + 6\right)\cdot 67^{6} + \left(15 a^{2} + 50 a + 61\right)\cdot 67^{7} + \left(15 a^{2} + 54 a + 60\right)\cdot 67^{8} + \left(6 a^{2} + 51 a + 24\right)\cdot 67^{9} + \left(61 a^{2} + 60 a + 43\right)\cdot 67^{10} + \left(32 a^{2} + 64\right)\cdot 67^{11} + \left(48 a^{2} + 26 a + 59\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a^{2} + 24 a + 62 + \left(53 a^{2} + 33 a + 13\right)\cdot 67 + \left(39 a^{2} + 4 a + 25\right)\cdot 67^{2} + \left(58 a^{2} + 39 a + 33\right)\cdot 67^{3} + \left(37 a^{2} + 16 a + 17\right)\cdot 67^{4} + \left(62 a^{2} + 32 a + 49\right)\cdot 67^{5} + \left(40 a^{2} + 11 a + 29\right)\cdot 67^{6} + \left(25 a^{2} + 65 a + 35\right)\cdot 67^{7} + \left(24 a^{2} + 22 a + 30\right)\cdot 67^{8} + \left(44 a^{2} + 11 a + 43\right)\cdot 67^{9} + \left(59 a^{2} + 41 a + 37\right)\cdot 67^{10} + \left(58 a^{2} + 4 a + 34\right)\cdot 67^{11} + \left(3 a^{2} + 30 a + 15\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a^{2} + 48 a + 38 + \left(a^{2} + 5 a + 6\right)\cdot 67 + \left(49 a^{2} + 47 a + 62\right)\cdot 67^{2} + \left(21 a^{2} + 18 a + 19\right)\cdot 67^{3} + \left(48 a^{2} + 2 a + 59\right)\cdot 67^{4} + \left(37 a^{2} + 9 a + 16\right)\cdot 67^{5} + \left(22 a^{2} + a + 23\right)\cdot 67^{6} + \left(37 a^{2} + 4 a + 15\right)\cdot 67^{7} + \left(38 a^{2} + 17 a + 20\right)\cdot 67^{8} + \left(9 a^{2} + 11 a + 38\right)\cdot 67^{9} + \left(39 a^{2} + 47 a + 22\right)\cdot 67^{10} + \left(8 a^{2} + 30 a + 34\right)\cdot 67^{11} + \left(29 a^{2} + 60 a + 49\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a^{2} + 58 a + 53 + \left(5 a^{2} + 19 a + 21\right)\cdot 67 + \left(28 a^{2} + 4 a + 45\right)\cdot 67^{2} + \left(37 a^{2} + 49 a + 15\right)\cdot 67^{3} + \left(64 a^{2} + 48 a + 57\right)\cdot 67^{4} + \left(63 a^{2} + 22 a + 54\right)\cdot 67^{5} + \left(16 a^{2} + 22 a\right)\cdot 67^{6} + \left(2 a^{2} + 34 a + 9\right)\cdot 67^{7} + \left(63 a^{2} + 19 a + 51\right)\cdot 67^{8} + \left(14 a^{2} + 9 a + 59\right)\cdot 67^{9} + \left(41 a^{2} + 57 a + 30\right)\cdot 67^{10} + \left(17 a^{2} + 37 a + 3\right)\cdot 67^{11} + \left(36 a^{2} + 8 a + 11\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 11\cdot 67 + 43\cdot 67^{2} + 24\cdot 67^{3} + 7\cdot 67^{4} + 53\cdot 67^{5} + 48\cdot 67^{6} + 39\cdot 67^{7} + 63\cdot 67^{8} + 49\cdot 67^{9} + 40\cdot 67^{10} + 13\cdot 67^{11} + 40\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 + 51\cdot 67 + 2\cdot 67^{2} + 16\cdot 67^{3} + 43\cdot 67^{4} + 24\cdot 67^{5} + 36\cdot 67^{6} + 39\cdot 67^{7} + 32\cdot 67^{8} + 35\cdot 67^{9} + 5\cdot 67^{10} + 67^{11} + 10\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 a^{2} + 52 a + 19 + \left(7 a^{2} + 13 a + 31\right)\cdot 67 + \left(66 a^{2} + 58 a + 63\right)\cdot 67^{2} + \left(37 a^{2} + 45 a + 17\right)\cdot 67^{3} + \left(31 a^{2} + a + 59\right)\cdot 67^{4} + \left(7 a^{2} + 12 a + 29\right)\cdot 67^{5} + \left(9 a^{2} + 33 a + 36\right)\cdot 67^{6} + \left(39 a^{2} + 34 a + 22\right)\cdot 67^{7} + \left(46 a^{2} + 24 a + 52\right)\cdot 67^{8} + \left(7 a^{2} + 46 a + 30\right)\cdot 67^{9} + \left(33 a^{2} + 35 a + 65\right)\cdot 67^{10} + \left(57 a^{2} + 24 a + 28\right)\cdot 67^{11} + \left(26 a^{2} + 28 a + 40\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 38 a^{2} + 46 a + 18 + \left(61 a^{2} + 15 a + 45\right)\cdot 67 + \left(58 a^{2} + 25 a + 34\right)\cdot 67^{2} + \left(48 a^{2} + 34 a + 61\right)\cdot 67^{3} + \left(17 a^{2} + 25 a + 3\right)\cdot 67^{4} + \left(2 a^{2} + 57 a + 9\right)\cdot 67^{5} + \left(26 a^{2} + 30 a + 37\right)\cdot 67^{6} + \left(14 a^{2} + 12 a + 57\right)\cdot 67^{7} + \left(13 a^{2} + 62 a + 52\right)\cdot 67^{8} + \left(51 a^{2} + 3 a + 3\right)\cdot 67^{9} + \left(33 a^{2} + 26 a + 1\right)\cdot 67^{10} + \left(25 a^{2} + 35 a + 35\right)\cdot 67^{11} + \left(56 a^{2} + 47 a + 24\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,7)(2,9,4)(3,5,8)$ |
| $(1,6)(2,4)(3,8)$ |
| $(1,8,4)(2,6,3)(5,9,7)$ |
| $(2,9,4)(3,8,5)$ |
| $(2,3)(4,8)(5,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,6)(2,4)(3,8)$ |
$0$ |
| $9$ |
$2$ |
$(2,3)(4,8)(5,9)$ |
$2$ |
| $9$ |
$2$ |
$(1,6)(2,8)(3,4)(5,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,7)(2,9,4)(3,5,8)$ |
$-3$ |
| $6$ |
$3$ |
$(1,8,4)(2,6,3)(5,9,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,7)(3,8,5)$ |
$0$ |
| $12$ |
$3$ |
$(1,5,4)(2,6,8)(3,9,7)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,4,6,8,2)(5,9,7)$ |
$0$ |
| $18$ |
$6$ |
$(1,6,7)(2,5,4,3,9,8)$ |
$-1$ |
| $18$ |
$6$ |
$(1,8,6,5,7,3)(4,9)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
