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:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 5 + \left(44 a + 18\right)\cdot 59 + 51\cdot 59^{2} + \left(35 a + 19\right)\cdot 59^{3} + \left(42 a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 27 + \left(48 a + 32\right)\cdot 59 + \left(47 a + 23\right)\cdot 59^{2} + \left(48 a + 38\right)\cdot 59^{3} + \left(53 a + 12\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 57 + \left(14 a + 10\right)\cdot 59 + \left(58 a + 7\right)\cdot 59^{2} + \left(23 a + 54\right)\cdot 59^{3} + \left(16 a + 8\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a + 45 + \left(10 a + 3\right)\cdot 59 + \left(11 a + 23\right)\cdot 59^{2} + \left(10 a + 39\right)\cdot 59^{3} + \left(5 a + 17\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 52\cdot 59 + 12\cdot 59^{2} + 25\cdot 59^{3} + 18\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.
