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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 15\cdot 61 + 30\cdot 61^{2} + 46\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 29 + \left(19 a + 58\right)\cdot 61 + \left(7 a + 46\right)\cdot 61^{2} + \left(41 a + 14\right)\cdot 61^{3} + \left(41 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 25 + \left(26 a + 39\right)\cdot 61 + \left(5 a + 57\right)\cdot 61^{2} + \left(29 a + 34\right)\cdot 61^{3} + \left(34 a + 22\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 48 + \left(41 a + 58\right)\cdot 61 + \left(53 a + 34\right)\cdot 61^{2} + \left(19 a + 48\right)\cdot 61^{3} + \left(19 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 44 + \left(34 a + 46\right)\cdot 61 + \left(55 a + 36\right)\cdot 61^{2} + \left(31 a + 58\right)\cdot 61^{3} + \left(26 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 25\cdot 61 + 37\cdot 61^{2} + 40\cdot 61^{3} + 53\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
