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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 13\cdot 67 + 27\cdot 67^{2} + 18\cdot 67^{3} + 54\cdot 67^{4} + 58\cdot 67^{5} + 61\cdot 67^{6} + 55\cdot 67^{7} + 35\cdot 67^{8} + 53\cdot 67^{9} + 35\cdot 67^{10} + 64\cdot 67^{11} +O\left(67^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 a^{2} + 28 a + 4 + \left(30 a^{2} + 6 a + 55\right)\cdot 67 + \left(42 a^{2} + 53 a + 44\right)\cdot 67^{2} + \left(53 a^{2} + 4 a + 31\right)\cdot 67^{3} + \left(19 a^{2} + 24 a + 13\right)\cdot 67^{4} + \left(39 a^{2} + 52 a + 18\right)\cdot 67^{5} + \left(45 a + 45\right)\cdot 67^{6} + \left(63 a^{2} + 58 a + 42\right)\cdot 67^{7} + \left(61 a^{2} + 24 a + 20\right)\cdot 67^{8} + \left(61 a^{2} + 10 a + 49\right)\cdot 67^{9} + \left(14 a^{2} + 13 a + 65\right)\cdot 67^{10} + \left(55 a^{2} + 25 a + 39\right)\cdot 67^{11} + \left(4 a^{2} + 20 a + 25\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 35 a + 47 + \left(30 a^{2} + 37 a + 43\right)\cdot 67 + \left(6 a^{2} + 3 a + 65\right)\cdot 67^{2} + \left(58 a^{2} + 35 a + 48\right)\cdot 67^{3} + \left(35 a^{2} + 14 a + 44\right)\cdot 67^{4} + \left(15 a^{2} + 32 a + 51\right)\cdot 67^{5} + \left(39 a^{2} + 32 a + 21\right)\cdot 67^{6} + \left(64 a^{2} + 34 a + 63\right)\cdot 67^{7} + \left(41 a^{2} + 29 a + 46\right)\cdot 67^{8} + \left(47 a^{2} + 40 a + 23\right)\cdot 67^{9} + \left(31 a^{2} + 56 a + 53\right)\cdot 67^{10} + \left(2 a^{2} + 57\right)\cdot 67^{11} + \left(27 a^{2} + 33 a + 55\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 21\cdot 67 + 27\cdot 67^{2} + 8\cdot 67^{3} + 37\cdot 67^{4} + 54\cdot 67^{5} + 13\cdot 67^{6} + 18\cdot 67^{7} + 3\cdot 67^{8} + 37\cdot 67^{9} + 32\cdot 67^{10} + 66\cdot 67^{11} + 2\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a^{2} + 49 a + 39 + \left(17 a^{2} + 19 a + 58\right)\cdot 67 + \left(53 a^{2} + 43 a + 51\right)\cdot 67^{2} + \left(60 a^{2} + 62 a + 59\right)\cdot 67^{3} + \left(61 a^{2} + 13 a + 14\right)\cdot 67^{4} + \left(6 a^{2} + 27 a + 17\right)\cdot 67^{5} + \left(5 a^{2} + 56 a + 19\right)\cdot 67^{6} + \left(14 a^{2} + 45 a + 62\right)\cdot 67^{7} + \left(25 a^{2} + 47 a + 46\right)\cdot 67^{8} + \left(42 a^{2} + 15 a + 2\right)\cdot 67^{9} + \left(47 a^{2} + 31 a + 50\right)\cdot 67^{10} + \left(50 a^{2} + 37 a + 49\right)\cdot 67^{11} + \left(25 a^{2} + a + 50\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 a^{2} + 37 a + 43 + \left(56 a^{2} + 9 a + 24\right)\cdot 67 + \left(35 a^{2} + 46 a + 18\right)\cdot 67^{2} + \left(3 a^{2} + 44 a + 32\right)\cdot 67^{3} + \left(23 a^{2} + 18 a + 26\right)\cdot 67^{4} + \left(10 a^{2} + 51 a + 36\right)\cdot 67^{5} + \left(6 a^{2} + 12 a\right)\cdot 67^{6} + \left(38 a^{2} + 45 a + 10\right)\cdot 67^{7} + \left(20 a^{2} + a + 56\right)\cdot 67^{8} + \left(65 a^{2} + 36 a + 62\right)\cdot 67^{9} + \left(17 a^{2} + 30 a + 10\right)\cdot 67^{10} + \left(66 a^{2} + 31 a + 17\right)\cdot 67^{11} + \left(45 a^{2} + 36 a + 56\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 49 a^{2} + 50 a + 2 + \left(19 a^{2} + 9 a + 2\right)\cdot 67 + \left(7 a^{2} + 20 a + 2\right)\cdot 67^{2} + \left(15 a^{2} + 36 a + 11\right)\cdot 67^{3} + \left(36 a^{2} + 38 a + 46\right)\cdot 67^{4} + \left(44 a^{2} + 7 a + 33\right)\cdot 67^{5} + \left(22 a^{2} + 45 a + 22\right)\cdot 67^{6} + \left(55 a^{2} + 53 a + 26\right)\cdot 67^{7} + \left(66 a^{2} + 56 a + 12\right)\cdot 67^{8} + \left(43 a^{2} + 10 a + 9\right)\cdot 67^{9} + \left(54 a^{2} + 46 a + 11\right)\cdot 67^{10} + \left(13 a^{2} + 28 a + 36\right)\cdot 67^{11} + \left(14 a^{2} + 32 a + 4\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 15 a^{2} + 2 a + 13 + \left(46 a^{2} + 51 a + 49\right)\cdot 67 + \left(55 a^{2} + 34 a + 30\right)\cdot 67^{2} + \left(9 a^{2} + 17 a + 57\right)\cdot 67^{3} + \left(24 a^{2} + 24 a + 30\right)\cdot 67^{4} + \left(17 a^{2} + 30 a + 64\right)\cdot 67^{5} + \left(60 a^{2} + 8 a + 15\right)\cdot 67^{6} + \left(32 a^{2} + 30 a + 56\right)\cdot 67^{7} + \left(51 a^{2} + 40 a + 45\right)\cdot 67^{8} + \left(6 a^{2} + 20 a + 29\right)\cdot 67^{9} + \left(34 a^{2} + 23 a + 8\right)\cdot 67^{10} + \left(12 a^{2} + 10 a + 3\right)\cdot 67^{11} + \left(16 a^{2} + 10 a + 4\right)\cdot 67^{12} +O\left(67^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,8,7,4)$ |
| $(5,6)(7,8)$ |
| $(5,7)(6,8)$ |
| $(1,2,8,6)(3,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$8$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-8$ |
| $6$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $6$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $6$ |
$2$ |
$(2,3)(6,7)$ |
$0$ |
| $12$ |
$2$ |
$(5,6)(7,8)$ |
$0$ |
| $12$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $32$ |
$3$ |
$(2,8,6)(3,5,7)$ |
$-1$ |
| $12$ |
$4$ |
$(1,8,4,5)(2,7,3,6)$ |
$0$ |
| $24$ |
$4$ |
$(1,2,5,6)(3,8,7,4)$ |
$0$ |
| $24$ |
$4$ |
$(1,3,5,6)(2,8,7,4)$ |
$0$ |
| $24$ |
$4$ |
$(1,4)(2,7,3,6)$ |
$0$ |
| $32$ |
$6$ |
$(1,4)(2,7,8,3,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
