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