Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 12\cdot 37 + 36\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 24 + \left(11 a + 30\right)\cdot 37 + \left(24 a + 12\right)\cdot 37^{2} + \left(20 a + 13\right)\cdot 37^{3} + \left(25 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 25 + \left(3 a + 8\right)\cdot 37 + \left(13 a + 12\right)\cdot 37^{2} + \left(8 a + 3\right)\cdot 37^{3} + \left(3 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 30 + \left(33 a + 31\right)\cdot 37 + \left(23 a + 23\right)\cdot 37^{2} + \left(28 a + 23\right)\cdot 37^{3} + \left(33 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 31 + \left(25 a + 27\right)\cdot 37 + \left(12 a + 24\right)\cdot 37^{2} + \left(16 a + 34\right)\cdot 37^{3} + \left(11 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
