Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 15\cdot 61 + 34\cdot 61^{2} + 54\cdot 61^{3} + 2\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 18 + \left(31 a + 8\right)\cdot 61 + \left(42 a + 38\right)\cdot 61^{2} + \left(40 a + 34\right)\cdot 61^{3} + \left(33 a + 32\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 34\cdot 61 + 59\cdot 61^{2} + 6\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 11 + \left(18 a + 36\right)\cdot 61 + \left(10 a + 4\right)\cdot 61^{2} + \left(53 a + 36\right)\cdot 61^{3} + \left(26 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 20 + \left(29 a + 37\right)\cdot 61 + \left(18 a + 49\right)\cdot 61^{2} + \left(20 a + 32\right)\cdot 61^{3} + \left(27 a + 25\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 14 + \left(42 a + 51\right)\cdot 61 + \left(50 a + 57\right)\cdot 61^{2} + \left(7 a + 17\right)\cdot 61^{3} + \left(34 a + 32\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(2,5)$ |
$0$ |
| $9$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$1$ |
| $4$ |
$3$ |
$(3,4,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,5,6)$ |
$1$ |
| $12$ |
$6$ |
$(2,5)(3,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
