Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 a + 50 + \left(35 a + 28\right)\cdot 59 + \left(49 a + 10\right)\cdot 59^{2} + \left(49 a + 53\right)\cdot 59^{3} + \left(17 a + 56\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 15\cdot 59 + 46\cdot 59^{2} + 53\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 47 + \left(52 a + 46\right)\cdot 59 + \left(45 a + 21\right)\cdot 59^{2} + \left(55 a + 3\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 48 + \left(23 a + 7\right)\cdot 59 + \left(9 a + 24\right)\cdot 59^{2} + \left(9 a + 53\right)\cdot 59^{3} + \left(41 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 9 + \left(6 a + 19\right)\cdot 59 + \left(13 a + 15\right)\cdot 59^{2} + \left(3 a + 13\right)\cdot 59^{3} + \left(a + 1\right)\cdot 59^{4} +O\left(59^{ 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.
