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:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 57 + \left(37 a + 10\right)\cdot 61 + \left(9 a + 57\right)\cdot 61^{2} + 33\cdot 61^{3} + \left(4 a + 4\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 a + 6 + \left(23 a + 38\right)\cdot 61 + \left(51 a + 29\right)\cdot 61^{2} + \left(60 a + 24\right)\cdot 61^{3} + \left(56 a + 8\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 11 + \left(53 a + 28\right)\cdot 61 + \left(30 a + 43\right)\cdot 61^{2} + \left(8 a + 37\right)\cdot 61^{3} + \left(21 a + 3\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 55\cdot 61 + 31\cdot 61^{2} + 10\cdot 61^{3} + 28\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 42 + \left(7 a + 50\right)\cdot 61 + \left(30 a + 20\right)\cdot 61^{2} + \left(52 a + 15\right)\cdot 61^{3} + \left(39 a + 16\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $10$ |
$2$ |
$(1,2)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
