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