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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 57\cdot 61 + 7\cdot 61^{2} + 35\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 33 + \left(2 a + 33\right)\cdot 61 + \left(9 a + 49\right)\cdot 61^{2} + \left(11 a + 14\right)\cdot 61^{3} + \left(13 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 a + 37 + \left(58 a + 31\right)\cdot 61 + \left(51 a + 56\right)\cdot 61^{2} + \left(49 a + 16\right)\cdot 61^{3} + \left(47 a + 29\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 26 + \left(46 a + 42\right)\cdot 61 + \left(42 a + 58\right)\cdot 61^{2} + \left(60 a + 3\right)\cdot 61^{3} + \left(56 a + 43\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 31 + \left(14 a + 22\right)\cdot 61 + \left(18 a + 55\right)\cdot 61^{2} + 21\cdot 61^{3} + \left(4 a + 39\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 56\cdot 61 + 15\cdot 61^{2} + 29\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(3,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,4,6,5,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
