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:
| $r_{ 1 }$ |
$=$ |
$ 57 a^{2} + 47 a + 27 + \left(46 a^{2} + 7 a + 53\right)\cdot 67 + \left(52 a^{2} + 20 a + 9\right)\cdot 67^{2} + \left(66 a^{2} + 6 a + 66\right)\cdot 67^{3} + \left(58 a^{2} + 58 a + 34\right)\cdot 67^{4} + \left(3 a^{2} + 59 a + 15\right)\cdot 67^{5} + \left(62 a^{2} + 15 a + 47\right)\cdot 67^{6} + \left(61 a^{2} + 17 a + 46\right)\cdot 67^{7} + \left(48 a^{2} + 9 a + 61\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 3\cdot 67 + 22\cdot 67^{3} + 31\cdot 67^{4} + 25\cdot 67^{5} + 39\cdot 67^{6} + 20\cdot 67^{7} + 60\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 a^{2} + 30 a + 14 + \left(65 a^{2} + 6 a + 61\right)\cdot 67 + \left(11 a^{2} + 52 a + 47\right)\cdot 67^{2} + \left(5 a^{2} + 62 a + 20\right)\cdot 67^{3} + \left(15 a^{2} + 13 a + 60\right)\cdot 67^{4} + \left(21 a^{2} + 21 a + 17\right)\cdot 67^{5} + \left(53 a^{2} + 21 a + 12\right)\cdot 67^{6} + \left(46 a^{2} + 52 a + 53\right)\cdot 67^{7} + \left(24 a^{2} + 50 a + 31\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a^{2} + 64 a + 61 + \left(34 a^{2} + 35 a + 3\right)\cdot 67 + \left(46 a^{2} + 27 a + 52\right)\cdot 67^{2} + \left(46 a^{2} + 23 a + 52\right)\cdot 67^{3} + \left(18 a^{2} + 6 a + 7\right)\cdot 67^{4} + \left(41 a^{2} + 18 a + 31\right)\cdot 67^{5} + \left(62 a^{2} + 37 a + 49\right)\cdot 67^{6} + \left(10 a^{2} + 40 a + 43\right)\cdot 67^{7} + \left(45 a^{2} + 5 a + 46\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 10\cdot 67 + 16\cdot 67^{2} + 21\cdot 67^{3} + 46\cdot 67^{4} + 39\cdot 67^{5} + 32\cdot 67^{6} + 55\cdot 67^{7} + 62\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a^{2} + 38 a + 49 + \left(8 a^{2} + 42 a + 33\right)\cdot 67 + \left(47 a^{2} + 57 a + 54\right)\cdot 67^{2} + \left(61 a^{2} + 29 a + 45\right)\cdot 67^{3} + \left(21 a^{2} + 38 a + 20\right)\cdot 67^{4} + \left(43 a^{2} + 35 a + 39\right)\cdot 67^{5} + \left(16 a^{2} + 32 a + 66\right)\cdot 67^{6} + \left(56 a^{2} + 37 a + 23\right)\cdot 67^{7} + \left(34 a^{2} + 60 a + 5\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 a^{2} + 23 a + 46 + \left(52 a^{2} + 23 a + 9\right)\cdot 67 + \left(34 a^{2} + 19 a + 5\right)\cdot 67^{2} + \left(20 a^{2} + 37 a + 15\right)\cdot 67^{3} + \left(56 a^{2} + 2 a + 24\right)\cdot 67^{4} + \left(21 a^{2} + 56 a + 20\right)\cdot 67^{5} + \left(9 a^{2} + 13 a + 37\right)\cdot 67^{6} + \left(61 a^{2} + 9 a + 43\right)\cdot 67^{7} + \left(39 a^{2} + 52 a + 25\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 62 + 52\cdot 67 + 50\cdot 67^{2} + 23\cdot 67^{3} + 56\cdot 67^{4} + 67^{5} + 62\cdot 67^{6} + 57\cdot 67^{7} + 10\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ a^{2} + 66 a + 4 + \left(60 a^{2} + 17 a + 39\right)\cdot 67 + \left(7 a^{2} + 24 a + 31\right)\cdot 67^{2} + 41 a\cdot 67^{3} + \left(30 a^{2} + 14 a + 53\right)\cdot 67^{4} + \left(2 a^{2} + 10 a + 9\right)\cdot 67^{5} + \left(64 a^{2} + 13 a + 55\right)\cdot 67^{6} + \left(30 a^{2} + 44 a + 56\right)\cdot 67^{7} + \left(7 a^{2} + 22 a + 29\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6)(3,8)(5,9)$ |
| $(1,4,7)(2,8,5)$ |
| $(1,3,8)(2,4,6)(5,7,9)$ |
| $(2,5)(4,7)(6,9)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,6)(3,8)(5,9)$ |
$2$ |
| $9$ |
$2$ |
$(2,5)(4,7)(6,9)$ |
$0$ |
| $9$ |
$2$ |
$(2,9)(3,8)(4,7)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,7,4)(2,8,5)(3,9,6)$ |
$-3$ |
| $6$ |
$3$ |
$(1,3,8)(2,4,6)(5,7,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,4,7)(2,8,5)$ |
$0$ |
| $12$ |
$3$ |
$(1,9,8)(2,4,3)(5,7,6)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,7)(2,6,8,3,5,9)$ |
$-1$ |
| $18$ |
$6$ |
$(1,3,8)(2,7,6,5,4,9)$ |
$0$ |
| $18$ |
$6$ |
$(1,4)(2,9,8,3,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
