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 a + 25 + \left(9 a + 54\right)\cdot 61 + \left(54 a + 43\right)\cdot 61^{2} + \left(19 a + 3\right)\cdot 61^{3} + \left(49 a + 59\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 9 + \left(40 a + 26\right)\cdot 61 + \left(55 a + 15\right)\cdot 61^{2} + \left(45 a + 48\right)\cdot 61^{3} + \left(44 a + 39\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 33 + \left(51 a + 55\right)\cdot 61 + \left(6 a + 27\right)\cdot 61^{2} + \left(41 a + 30\right)\cdot 61^{3} + \left(11 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 21 + \left(20 a + 54\right)\cdot 61 + \left(5 a + 30\right)\cdot 61^{2} + \left(15 a + 38\right)\cdot 61^{3} + \left(16 a + 38\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 53\cdot 61 + 3\cdot 61^{2} + 61^{3} + 18\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
