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:
| $r_{ 1 }$ |
$=$ |
$ 35 + 19\cdot 61 + 2\cdot 61^{2} + 2\cdot 61^{3} + 33\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 47 + \left(44 a + 16\right)\cdot 61 + \left(a + 56\right)\cdot 61^{2} + \left(8 a + 15\right)\cdot 61^{3} + \left(35 a + 57\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 11 + \left(30 a + 24\right)\cdot 61 + \left(48 a + 56\right)\cdot 61^{2} + \left(57 a + 50\right)\cdot 61^{3} + \left(14 a + 5\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 54 + \left(30 a + 11\right)\cdot 61 + \left(12 a + 13\right)\cdot 61^{2} + \left(3 a + 60\right)\cdot 61^{3} + \left(46 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 41\cdot 61^{2} + 31\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 59 + \left(16 a + 48\right)\cdot 61 + \left(59 a + 13\right)\cdot 61^{2} + \left(52 a + 22\right)\cdot 61^{3} + \left(25 a + 23\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,4)$ |
$-2$ |
| $9$ |
$2$ |
$(3,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $18$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,3,6,4,2)$ |
$0$ |
| $12$ |
$6$ |
$(2,5,6)(3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
