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