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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 33 + \left(29 a + 13\right)\cdot 37 + \left(33 a + 36\right)\cdot 37^{2} + \left(29 a + 35\right)\cdot 37^{3} + \left(18 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 5\cdot 37 + 8\cdot 37^{2} + 34\cdot 37^{3} + 23\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 22 + \left(19 a + 6\right)\cdot 37 + \left(28 a + 11\right)\cdot 37^{2} + \left(25 a + 6\right)\cdot 37^{3} + \left(8 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 20\cdot 37 + 29\cdot 37^{2} + 16\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 33 + \left(17 a + 34\right)\cdot 37 + \left(8 a + 31\right)\cdot 37^{2} + \left(11 a + 6\right)\cdot 37^{3} + \left(28 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 5 + \left(7 a + 29\right)\cdot 37 + \left(3 a + 30\right)\cdot 37^{2} + \left(7 a + 10\right)\cdot 37^{3} + \left(18 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$2$ |
| $6$ |
$2$ |
$(2,6)$ |
$0$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,5,6,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,5,6,3)$ |
$-1$ |
| $12$ |
$6$ |
$(2,6)(3,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
