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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 12 + 14\cdot 37 + \left(25 a + 32\right)\cdot 37^{2} + \left(29 a + 21\right)\cdot 37^{3} + \left(3 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 14\cdot 37 + 14\cdot 37^{2} + 25\cdot 37^{3} + 8\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 1 + \left(28 a + 23\right)\cdot 37 + \left(13 a + 1\right)\cdot 37^{2} + \left(30 a + 18\right)\cdot 37^{3} + \left(11 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 3\cdot 37 + 14\cdot 37^{2} + 26\cdot 37^{3} + 33\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 33 + \left(36 a + 21\right)\cdot 37 + \left(11 a + 20\right)\cdot 37^{2} + \left(7 a + 4\right)\cdot 37^{3} + \left(33 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 14 + \left(8 a + 33\right)\cdot 37 + \left(23 a + 27\right)\cdot 37^{2} + \left(6 a + 14\right)\cdot 37^{3} + \left(25 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $6$ |
$2$ |
$(3,6)$ |
$0$ |
| $9$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$1$ |
| $4$ |
$3$ |
$(1,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,4,6,5,2)$ |
$-1$ |
| $12$ |
$6$ |
$(1,4,5)(3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
