Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 a + 18 + \left(25 a + 4\right)\cdot 37 + \left(23 a + 27\right)\cdot 37^{2} + \left(23 a + 32\right)\cdot 37^{3} + \left(6 a + 34\right)\cdot 37^{4} + \left(10 a + 30\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 6 + 11 a\cdot 37 + \left(13 a + 22\right)\cdot 37^{2} + \left(13 a + 29\right)\cdot 37^{3} + 30 a\cdot 37^{4} + \left(26 a + 28\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 7\cdot 37 + 5\cdot 37^{2} + 4\cdot 37^{3} + 2\cdot 37^{4} + 11\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 19 + \left(11 a + 32\right)\cdot 37 + \left(13 a + 9\right)\cdot 37^{2} + \left(13 a + 4\right)\cdot 37^{3} + \left(30 a + 2\right)\cdot 37^{4} + \left(26 a + 6\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 31 + \left(25 a + 36\right)\cdot 37 + \left(23 a + 14\right)\cdot 37^{2} + \left(23 a + 7\right)\cdot 37^{3} + \left(6 a + 36\right)\cdot 37^{4} + \left(10 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 29\cdot 37 + 31\cdot 37^{2} + 32\cdot 37^{3} + 34\cdot 37^{4} + 25\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,2,4,6,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
