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