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