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