Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 a^{2} + 4 a + 51 + \left(61 a^{2} + 65 a + 20\right)\cdot 67 + \left(13 a^{2} + 58 a + 42\right)\cdot 67^{2} + \left(46 a^{2} + 55 a + 38\right)\cdot 67^{3} + \left(27 a^{2} + 4 a + 60\right)\cdot 67^{4} + \left(53 a^{2} + 2 a + 15\right)\cdot 67^{5} + \left(53 a^{2} + a + 48\right)\cdot 67^{6} + \left(33 a^{2} + 34\right)\cdot 67^{7} + \left(5 a^{2} + 11 a + 42\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 45\cdot 67 + 2\cdot 67^{2} + 13\cdot 67^{3} + 19\cdot 67^{4} + 35\cdot 67^{5} + 57\cdot 67^{6} + 8\cdot 67^{7} + 7\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 36\cdot 67 + 2\cdot 67^{2} + 3\cdot 67^{3} + 45\cdot 67^{4} + 42\cdot 67^{5} + 19\cdot 67^{6} + 6\cdot 67^{7} + 3\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 a^{2} + 50 a + \left(16 a^{2} + 44 a + 38\right)\cdot 67 + \left(66 a^{2} + 45 a + 64\right)\cdot 67^{2} + \left(19 a^{2} + 11 a + 19\right)\cdot 67^{3} + \left(a^{2} + 17 a + 12\right)\cdot 67^{4} + \left(48 a^{2} + 40 a + 3\right)\cdot 67^{5} + \left(21 a^{2} + 36 a + 59\right)\cdot 67^{6} + \left(47 a^{2} + 28 a + 34\right)\cdot 67^{7} + \left(36 a^{2} + 65 a + 56\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a^{2} + 16 a + 58 + \left(33 a^{2} + 42 a + 37\right)\cdot 67 + \left(14 a^{2} + 30 a + 58\right)\cdot 67^{2} + \left(62 a^{2} + 61 a + 54\right)\cdot 67^{3} + \left(65 a^{2} + 10 a + 2\right)\cdot 67^{4} + \left(30 a^{2} + 5 a + 2\right)\cdot 67^{5} + \left(37 a^{2} + 7 a + 55\right)\cdot 67^{6} + \left(2 a^{2} + 26 a + 56\right)\cdot 67^{7} + \left(50 a^{2} + 8 a + 42\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 11\cdot 67 + 33\cdot 67^{2} + 66\cdot 67^{3} + 64\cdot 67^{4} + 9\cdot 67^{5} + 38\cdot 67^{6} + 12\cdot 67^{7} + 65\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 a^{2} + a + \left(16 a^{2} + 47 a + 38\right)\cdot 67 + \left(53 a^{2} + 57 a + 12\right)\cdot 67^{2} + \left(51 a^{2} + 60 a + 13\right)\cdot 67^{3} + \left(66 a^{2} + 38 a + 6\right)\cdot 67^{4} + \left(54 a^{2} + 21 a + 31\right)\cdot 67^{5} + \left(7 a^{2} + 23 a + 3\right)\cdot 67^{6} + \left(17 a^{2} + 12 a + 48\right)\cdot 67^{7} + \left(47 a^{2} + 60 a + 31\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 a^{2} + 6 a + 21 + \left(14 a^{2} + 50 a + 35\right)\cdot 67 + \left(55 a^{2} + 26 a + 6\right)\cdot 67^{2} + \left(23 a^{2} + 17 a + 16\right)\cdot 67^{3} + \left(52 a^{2} + 30 a + 25\right)\cdot 67^{4} + \left(25 a^{2} + 3 a + 39\right)\cdot 67^{5} + \left(17 a^{2} + 33 a + 36\right)\cdot 67^{6} + \left(40 a^{2} + 44 a + 60\right)\cdot 67^{7} + \left(63 a^{2} + 23 a + 6\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 40 a^{2} + 57 a + 8 + \left(57 a^{2} + 18 a + 5\right)\cdot 67 + \left(64 a^{2} + 48 a + 45\right)\cdot 67^{2} + \left(63 a^{2} + 60 a + 42\right)\cdot 67^{3} + \left(53 a^{2} + 31 a + 31\right)\cdot 67^{4} + \left(54 a^{2} + 61 a + 21\right)\cdot 67^{5} + \left(62 a^{2} + 32 a + 17\right)\cdot 67^{6} + \left(59 a^{2} + 22 a + 5\right)\cdot 67^{7} + \left(64 a^{2} + 32 a + 12\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6,3)(4,5,7)$ |
| $(2,5)(3,4)(6,7)(8,9)$ |
| $(1,2,4,9,6,7,8,3,5)$ |
| $(1,8,9)(4,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,5)(3,4)(6,7)(8,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,8)(2,6,3)(4,7,5)$ |
$-3$ |
| $3$ |
$3$ |
$(1,8,9)(4,7,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,9,8)(4,5,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,8)(2,4,3,7,6,5)$ |
$0$ |
| $9$ |
$6$ |
$(1,8)(2,5,6,7,3,4)$ |
$0$ |
| $6$ |
$9$ |
$(1,2,4,9,6,7,8,3,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,6,4,9,3,7,8,2,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,4,3,8,5,6,9,7,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
