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