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:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 22 + \left(17 a + 6\right)\cdot 37 + \left(2 a + 36\right)\cdot 37^{2} + \left(a + 13\right)\cdot 37^{3} + \left(27 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 2 + \left(19 a + 9\right)\cdot 37 + \left(34 a + 28\right)\cdot 37^{2} + \left(35 a + 15\right)\cdot 37^{3} + \left(9 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 35\cdot 37 + 23\cdot 37^{2} + 8\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + \left(12 a + 18\right)\cdot 37 + \left(25 a + 17\right)\cdot 37^{2} + \left(29 a + 4\right)\cdot 37^{3} + \left(20 a + 28\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 3 + \left(24 a + 20\right)\cdot 37 + \left(11 a + 32\right)\cdot 37^{2} + \left(7 a + 23\right)\cdot 37^{3} + \left(16 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 21\cdot 37 + 9\cdot 37^{2} + 7\cdot 37^{3} + 19\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(4,5)$ |
$-2$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(3,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,6)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
