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:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 29 + \left(35 a + 8\right)\cdot 37 + \left(20 a + 9\right)\cdot 37^{2} + \left(8 a + 35\right)\cdot 37^{3} + \left(a + 14\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 5\cdot 37 + 7\cdot 37^{2} + 27\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 5 + \left(33 a + 34\right)\cdot 37 + \left(2 a + 31\right)\cdot 37^{2} + \left(a + 32\right)\cdot 37^{3} + \left(2 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 29\cdot 37 + 31\cdot 37^{2} + 6\cdot 37^{3} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 23 + \left(a + 22\right)\cdot 37 + \left(16 a + 20\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(35 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 4 + \left(3 a + 10\right)\cdot 37 + \left(34 a + 10\right)\cdot 37^{2} + \left(35 a + 34\right)\cdot 37^{3} + \left(34 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(2,5)$ |
$0$ |
| $9$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,6,5,3)$ |
$1$ |
| $12$ |
$6$ |
$(2,5)(3,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
