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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 36 + \left(33 a + 42\right)\cdot 59 + \left(37 a + 20\right)\cdot 59^{2} + \left(50 a + 24\right)\cdot 59^{3} + \left(19 a + 47\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 24\cdot 59 + 47\cdot 59^{2} + 17\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 27 + \left(50 a + 10\right)\cdot 59 + \left(22 a + 26\right)\cdot 59^{2} + \left(30 a + 15\right)\cdot 59^{3} + \left(34 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 58 + \left(8 a + 29\right)\cdot 59 + \left(36 a + 57\right)\cdot 59^{2} + \left(28 a + 22\right)\cdot 59^{3} + \left(24 a + 10\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 a + 42 + \left(25 a + 10\right)\cdot 59 + \left(21 a + 25\right)\cdot 59^{2} + \left(8 a + 37\right)\cdot 59^{3} + \left(39 a + 16\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.
