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