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