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:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 30 + \left(34 a + 56\right)\cdot 61 + \left(40 a + 10\right)\cdot 61^{2} + \left(4 a + 46\right)\cdot 61^{3} + \left(17 a + 18\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 19 + \left(26 a + 41\right)\cdot 61 + \left(20 a + 16\right)\cdot 61^{2} + \left(56 a + 10\right)\cdot 61^{3} + \left(43 a + 31\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 26\cdot 61 + 38\cdot 61^{2} + 57\cdot 61^{3} + 49\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 24\cdot 61 + 33\cdot 61^{2} + 4\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 10 + \left(30 a + 60\right)\cdot 61 + 51 a\cdot 61^{2} + \left(30 a + 12\right)\cdot 61^{3} + \left(47 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 5 + \left(30 a + 35\right)\cdot 61 + \left(9 a + 21\right)\cdot 61^{2} + \left(30 a + 52\right)\cdot 61^{3} + \left(13 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$2$ |
| $6$ |
$2$ |
$(2,4)$ |
$0$ |
| $9$ |
$2$ |
$(2,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$1$ |
| $4$ |
$3$ |
$(3,5,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,5,4,6)$ |
$-1$ |
| $12$ |
$6$ |
$(2,4)(3,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
