Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 38\cdot 61 + 45\cdot 61^{2} + 23\cdot 61^{3} + 49\cdot 61^{4} + 51\cdot 61^{5} + 37\cdot 61^{6} + 26\cdot 61^{7} + 45\cdot 61^{8} + 36\cdot 61^{9} + 43\cdot 61^{10} + 20\cdot 61^{11} + 14\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a^{2} + 25 a + 26 + \left(35 a^{2} + 24 a + 3\right)\cdot 61 + \left(6 a^{2} + 54 a + 51\right)\cdot 61^{2} + \left(20 a^{2} + 49 a + 32\right)\cdot 61^{3} + \left(32 a^{2} + 12 a + 8\right)\cdot 61^{4} + \left(36 a^{2} + 55 a + 28\right)\cdot 61^{5} + \left(54 a^{2} + 51 a + 51\right)\cdot 61^{6} + \left(53 a^{2} + 29 a + 27\right)\cdot 61^{7} + \left(48 a^{2} + 58 a + 4\right)\cdot 61^{8} + \left(24 a^{2} + 50 a + 14\right)\cdot 61^{9} + \left(11 a^{2} + 59 a + 53\right)\cdot 61^{10} + \left(55 a^{2} + 27 a + 33\right)\cdot 61^{11} + \left(4 a^{2} + 47 a + 2\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 26\cdot 61^{2} + 19\cdot 61^{3} + 9\cdot 61^{4} + 56\cdot 61^{5} + 61^{6} + 39\cdot 61^{7} + 54\cdot 61^{8} + 45\cdot 61^{9} + 17\cdot 61^{10} + 30\cdot 61^{11} + 16\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 22\cdot 61 + 50\cdot 61^{2} + 17\cdot 61^{3} + 2\cdot 61^{4} + 14\cdot 61^{5} + 21\cdot 61^{6} + 56\cdot 61^{7} + 21\cdot 61^{8} + 39\cdot 61^{9} + 60\cdot 61^{10} + 9\cdot 61^{11} + 30\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 13 a + 5 + \left(57 a^{2} + 58 a + 22\right)\cdot 61 + \left(7 a^{2} + 35 a + 57\right)\cdot 61^{2} + \left(12 a^{2} + a + 15\right)\cdot 61^{3} + \left(11 a^{2} + 20 a + 52\right)\cdot 61^{4} + \left(11 a^{2} + 4 a + 31\right)\cdot 61^{5} + \left(22 a^{2} + 35 a + 42\right)\cdot 61^{6} + \left(23 a^{2} + 14 a + 27\right)\cdot 61^{7} + \left(46 a^{2} + 30 a + 33\right)\cdot 61^{8} + \left(40 a^{2} + 15 a + 27\right)\cdot 61^{9} + \left(29 a^{2} + 38 a + 16\right)\cdot 61^{10} + \left(46 a^{2} + 24 a + 54\right)\cdot 61^{11} + \left(12 a^{2} + 10 a + 18\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a^{2} + 23 a + 59 + \left(49 a^{2} + 56 a + 47\right)\cdot 61 + \left(12 a^{2} + 53 a + 59\right)\cdot 61^{2} + \left(40 a^{2} + 12 a + 24\right)\cdot 61^{3} + \left(31 a^{2} + 52 a + 5\right)\cdot 61^{4} + \left(51 a^{2} + 38 a + 37\right)\cdot 61^{5} + \left(20 a^{2} + 2 a + 56\right)\cdot 61^{6} + \left(40 a^{2} + 25 a + 45\right)\cdot 61^{7} + \left(6 a^{2} + 60 a + 10\right)\cdot 61^{8} + \left(49 a^{2} + 26 a + 46\right)\cdot 61^{9} + \left(8 a^{2} + 56 a + 20\right)\cdot 61^{10} + \left(28 a^{2} + 52 a + 9\right)\cdot 61^{11} + \left(45 a^{2} + 39 a + 29\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 a^{2} + 23 a + 31 + \left(29 a^{2} + 39 a + 35\right)\cdot 61 + \left(46 a^{2} + 31 a + 13\right)\cdot 61^{2} + \left(28 a^{2} + 9 a + 12\right)\cdot 61^{3} + \left(17 a^{2} + 28 a\right)\cdot 61^{4} + \left(13 a^{2} + a + 1\right)\cdot 61^{5} + \left(45 a^{2} + 35 a + 28\right)\cdot 61^{6} + \left(44 a^{2} + 16 a + 5\right)\cdot 61^{7} + \left(26 a^{2} + 33 a + 23\right)\cdot 61^{8} + \left(56 a^{2} + 55 a + 19\right)\cdot 61^{9} + \left(19 a^{2} + 23 a + 52\right)\cdot 61^{10} + \left(20 a^{2} + 8 a + 33\right)\cdot 61^{11} + \left(43 a^{2} + 3 a + 39\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 37 a^{2} + 10 a + 51 + \left(49 a^{2} + 48 a + 7\right)\cdot 61 + \left(8 a^{2} + 16 a + 41\right)\cdot 61^{2} + \left(40 a^{2} + a + 24\right)\cdot 61^{3} + \left(17 a^{2} + 49 a + 21\right)\cdot 61^{4} + \left(42 a^{2} + 40 a + 14\right)\cdot 61^{5} + \left(34 a^{2} + 12 a + 60\right)\cdot 61^{6} + \left(34 a^{2} + 53 a + 18\right)\cdot 61^{7} + \left(20 a^{2} + a + 55\right)\cdot 61^{8} + \left(28 a^{2} + 55 a + 50\right)\cdot 61^{9} + \left(21 a^{2} + 21 a + 18\right)\cdot 61^{10} + \left(52 a^{2} + 3 a\right)\cdot 61^{11} + \left(3 a^{2} + 15 a + 18\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 55 a^{2} + 28 a + 13 + \left(22 a^{2} + 17 a + 5\right)\cdot 61 + \left(39 a^{2} + 51 a + 21\right)\cdot 61^{2} + \left(41 a^{2} + 46 a + 11\right)\cdot 61^{3} + \left(11 a^{2} + 20 a + 34\right)\cdot 61^{4} + \left(28 a^{2} + 42 a + 9\right)\cdot 61^{5} + \left(5 a^{2} + 45 a + 5\right)\cdot 61^{6} + \left(47 a^{2} + 43 a + 57\right)\cdot 61^{7} + \left(33 a^{2} + 59 a + 55\right)\cdot 61^{8} + \left(44 a^{2} + 39 a + 24\right)\cdot 61^{9} + \left(30 a^{2} + 43 a + 21\right)\cdot 61^{10} + \left(41 a^{2} + 4 a + 51\right)\cdot 61^{11} + \left(11 a^{2} + 6 a + 13\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,3)(2,7,5)(6,8,9)$ |
| $(1,3,4)(6,8,9)$ |
| $(1,9)(2,5)(3,6)(4,8)$ |
| $(1,6,2)(3,9,5)(4,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,5)(3,6)(4,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,3)(2,7,5)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,3,4)(2,7,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,3)(2,5,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,2)(3,9,5)(4,8,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,7)(2,3,6)(4,9,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,8)(2,6,3)(4,5,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,5,3,2,4,7)(8,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,7,4,2,3,5)(8,9)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
