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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 16 + \left(53 a + 50\right)\cdot 61 + \left(13 a + 11\right)\cdot 61^{2} + \left(54 a + 13\right)\cdot 61^{3} + \left(6 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 46 + \left(45 a + 9\right)\cdot 61 + \left(44 a + 6\right)\cdot 61^{2} + \left(6 a + 7\right)\cdot 61^{3} + \left(18 a + 24\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 27\cdot 61 + 4\cdot 61^{2} + 18\cdot 61^{3} + 22\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 49 + \left(7 a + 9\right)\cdot 61 + \left(47 a + 33\right)\cdot 61^{2} + \left(6 a + 53\right)\cdot 61^{3} + \left(54 a + 26\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 16 + \left(15 a + 24\right)\cdot 61 + \left(16 a + 5\right)\cdot 61^{2} + \left(54 a + 30\right)\cdot 61^{3} + \left(42 a + 35\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
