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