Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{3} + 6 x + 65 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 26\cdot 67 + 17\cdot 67^{2} + 36\cdot 67^{3} + 5\cdot 67^{4} + 2\cdot 67^{5} + 50\cdot 67^{6} + 19\cdot 67^{7} + 17\cdot 67^{8} + 28\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a^{2} + 60 a + 28 + \left(60 a^{2} + 16 a + 39\right)\cdot 67 + \left(32 a^{2} + 51 a + 64\right)\cdot 67^{2} + \left(45 a^{2} + 20 a + 47\right)\cdot 67^{3} + \left(17 a^{2} + 11 a + 3\right)\cdot 67^{4} + \left(53 a^{2} + 22 a + 12\right)\cdot 67^{5} + \left(19 a^{2} + 39 a + 12\right)\cdot 67^{6} + \left(65 a^{2} + 36 a + 60\right)\cdot 67^{7} + \left(29 a^{2} + 4 a + 52\right)\cdot 67^{8} + \left(58 a^{2} + 47 a + 32\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a^{2} + 32 a + 64 + \left(64 a^{2} + 11 a + 55\right)\cdot 67 + \left(23 a^{2} + 44 a + 28\right)\cdot 67^{2} + \left(52 a^{2} + 30 a + 8\right)\cdot 67^{3} + \left(29 a^{2} + 51 a + 52\right)\cdot 67^{4} + \left(32 a^{2} + 49 a + 62\right)\cdot 67^{5} + \left(10 a^{2} + 14 a + 41\right)\cdot 67^{6} + \left(8 a^{2} + 46 a + 32\right)\cdot 67^{7} + \left(55 a^{2} + 65 a + 19\right)\cdot 67^{8} + \left(44 a^{2} + 38 a + 45\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 24\cdot 67 + 50\cdot 67^{2} + 63\cdot 67^{3} + 48\cdot 67^{4} + 56\cdot 67^{5} + 38\cdot 67^{6} + 13\cdot 67^{7} + 12\cdot 67^{8} + 13\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a^{2} + 9 a + 31 + \left(2 a^{2} + 32 a + 11\right)\cdot 67 + 55 a\cdot 67^{2} + \left(6 a^{2} + 26 a + 24\right)\cdot 67^{3} + 43 a\cdot 67^{4} + \left(28 a^{2} + 47 a + 45\right)\cdot 67^{5} + \left(3 a^{2} + 17 a + 13\right)\cdot 67^{6} + \left(53 a^{2} + 59 a + 11\right)\cdot 67^{7} + \left(44 a^{2} + 20 a + 45\right)\cdot 67^{8} + \left(21 a^{2} + 48 a + 19\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 a^{2} + 45 a + 23 + \left(56 a^{2} + 5 a + 26\right)\cdot 67 + \left(28 a^{2} + 48\right)\cdot 67^{2} + \left(8 a^{2} + 26 a + 33\right)\cdot 67^{3} + \left(61 a^{2} + 66 a + 43\right)\cdot 67^{4} + \left(56 a^{2} + 18 a + 26\right)\cdot 67^{5} + \left(50 a^{2} + 55 a + 2\right)\cdot 67^{6} + \left(51 a^{2} + 59 a + 6\right)\cdot 67^{7} + \left(14 a^{2} + 32 a + 59\right)\cdot 67^{8} + \left(57 a^{2} + 27\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 15\cdot 67 + 66\cdot 67^{2} + 33\cdot 67^{3} + 12\cdot 67^{4} + 8\cdot 67^{5} + 45\cdot 67^{6} + 33\cdot 67^{7} + 37\cdot 67^{8} + 25\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 2 a^{2} + 65 a + 8 + \left(4 a^{2} + 17 a + 16\right)\cdot 67 + \left(34 a^{2} + 27 a + 2\right)\cdot 67^{2} + \left(15 a^{2} + 19 a + 62\right)\cdot 67^{3} + \left(49 a^{2} + 12 a + 62\right)\cdot 67^{4} + \left(52 a^{2} + 64 a + 9\right)\cdot 67^{5} + \left(43 a^{2} + 9 a + 41\right)\cdot 67^{6} + \left(15 a^{2} + 38 a + 62\right)\cdot 67^{7} + \left(59 a^{2} + 41 a + 35\right)\cdot 67^{8} + \left(53 a^{2} + 38 a + 14\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 62 a^{2} + 57 a + 47 + \left(12 a^{2} + 49 a + 51\right)\cdot 67 + \left(14 a^{2} + 22 a + 56\right)\cdot 67^{2} + \left(6 a^{2} + 10 a + 24\right)\cdot 67^{3} + \left(43 a^{2} + 16 a + 38\right)\cdot 67^{4} + \left(44 a^{2} + 65 a + 44\right)\cdot 67^{5} + \left(5 a^{2} + 63 a + 22\right)\cdot 67^{6} + \left(7 a^{2} + 27 a + 28\right)\cdot 67^{7} + \left(64 a^{2} + 35 a + 55\right)\cdot 67^{8} + \left(31 a^{2} + 27 a + 60\right)\cdot 67^{9} +O\left(67^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,7)(2,8,5)$ |
| $(2,8)(4,7)(6,9)$ |
| $(1,3,5)(2,7,9)(4,6,8)$ |
| $(2,9)(3,5)(6,8)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,9)(3,5)(6,8)$ |
$2$ |
| $9$ |
$2$ |
$(2,8)(4,7)(6,9)$ |
$0$ |
| $9$ |
$2$ |
$(2,6)(3,5)(4,7)(8,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,7,4)(2,8,5)(3,9,6)$ |
$-3$ |
| $6$ |
$3$ |
$(1,3,5)(2,7,9)(4,6,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,4,7)(2,8,5)$ |
$0$ |
| $12$ |
$3$ |
$(1,9,5)(2,7,6)(3,8,4)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,7)(2,9,8,6,5,3)$ |
$-1$ |
| $18$ |
$6$ |
$(1,3,5)(2,4,9,8,7,6)$ |
$0$ |
| $18$ |
$6$ |
$(1,4)(2,6,8,9,5,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
