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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 35 + 14 a\cdot 37 + \left(28 a + 6\right)\cdot 37^{2} + \left(32 a + 29\right)\cdot 37^{3} + \left(5 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 27\cdot 37 + 14\cdot 37^{2} + 26\cdot 37^{3} + 24\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 15 + \left(a + 12\right)\cdot 37 + \left(22 a + 23\right)\cdot 37^{2} + \left(33 a + 4\right)\cdot 37^{3} + \left(19 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 21 + \left(22 a + 6\right)\cdot 37 + \left(8 a + 31\right)\cdot 37^{2} + \left(4 a + 20\right)\cdot 37^{3} + \left(31 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 29 + \left(35 a + 33\right)\cdot 37 + \left(14 a + 35\right)\cdot 37^{2} + \left(3 a + 5\right)\cdot 37^{3} + \left(17 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 29\cdot 37 + 36\cdot 37^{2} + 23\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(2,3)$ |
$0$ |
| $9$ |
$2$ |
$(1,4)(2,3)$ |
$0$ |
| $4$ |
$3$ |
$(2,3,5)$ |
$-2$ |
| $4$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$1$ |
| $18$ |
$4$ |
$(1,2,4,3)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,4,3,6,5)$ |
$1$ |
| $12$ |
$6$ |
$(1,4,6)(2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
