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