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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 33 + \left(13 a + 26\right)\cdot 37 + \left(17 a + 23\right)\cdot 37^{2} + \left(35 a + 25\right)\cdot 37^{3} + \left(34 a + 20\right)\cdot 37^{4} + 8 a\cdot 37^{5} + \left(32 a + 28\right)\cdot 37^{6} + \left(34 a + 31\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 35\cdot 37 + 13\cdot 37^{2} + 26\cdot 37^{3} + 33\cdot 37^{4} + 11\cdot 37^{5} + 30\cdot 37^{6} + 8\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 10 + \left(23 a + 22\right)\cdot 37 + \left(19 a + 5\right)\cdot 37^{2} + \left(a + 2\right)\cdot 37^{3} + \left(2 a + 14\right)\cdot 37^{4} + \left(28 a + 1\right)\cdot 37^{5} + 4 a\cdot 37^{6} + \left(2 a + 28\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 19\cdot 37 + 7\cdot 37^{2} + 31\cdot 37^{4} + 12\cdot 37^{5} + 5\cdot 37^{6} + 32\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + \left(28 a + 30\right)\cdot 37 + \left(34 a + 11\right)\cdot 37^{2} + \left(35 a + 29\right)\cdot 37^{3} + \left(20 a + 18\right)\cdot 37^{4} + \left(22 a + 7\right)\cdot 37^{5} + \left(29 a + 31\right)\cdot 37^{6} + \left(4 a + 28\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a + 2 + \left(8 a + 14\right)\cdot 37 + \left(2 a + 11\right)\cdot 37^{2} + \left(a + 27\right)\cdot 37^{3} + \left(16 a + 29\right)\cdot 37^{4} + \left(14 a + 2\right)\cdot 37^{5} + \left(7 a + 16\right)\cdot 37^{6} + \left(32 a + 18\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,5)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(2,4)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,5,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,5)(2,6,4,3)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,6,5,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
