Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 28 + \left(16 a + 1\right)\cdot 37 + \left(26 a + 1\right)\cdot 37^{2} + \left(30 a + 9\right)\cdot 37^{3} + \left(8 a + 7\right)\cdot 37^{4} + \left(34 a + 32\right)\cdot 37^{5} + \left(19 a + 14\right)\cdot 37^{6} + \left(25 a + 19\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 17 + \left(20 a + 6\right)\cdot 37 + \left(10 a + 16\right)\cdot 37^{2} + \left(6 a + 31\right)\cdot 37^{3} + \left(28 a + 11\right)\cdot 37^{4} + \left(2 a + 12\right)\cdot 37^{5} + \left(17 a + 23\right)\cdot 37^{6} + \left(11 a + 27\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 8 + \left(35 a + 13\right)\cdot 37 + \left(28 a + 32\right)\cdot 37^{2} + \left(34 a + 34\right)\cdot 37^{3} + \left(26 a + 18\right)\cdot 37^{4} + \left(26 a + 29\right)\cdot 37^{5} + \left(23 a + 34\right)\cdot 37^{6} + \left(29 a + 2\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 18\cdot 37 + 27\cdot 37^{2} + 12\cdot 37^{3} + 24\cdot 37^{4} + 9\cdot 37^{5} + 24\cdot 37^{6} + 4\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 15\cdot 37 + 32\cdot 37^{2} + 25\cdot 37^{3} + 30\cdot 37^{4} + 28\cdot 37^{5} + 21\cdot 37^{6} + 32\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 5 + \left(a + 18\right)\cdot 37 + \left(8 a + 1\right)\cdot 37^{2} + \left(2 a + 34\right)\cdot 37^{3} + \left(10 a + 17\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + \left(13 a + 28\right)\cdot 37^{6} + \left(7 a + 23\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,6)$ |
| $(1,4,3)(2,5,6)$ |
| $(1,3)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,2,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$1$ |
| $8$ |
$6$ |
$(1,4,3,2,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
