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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 18 + \left(34 a + 21\right)\cdot 37 + \left(8 a + 24\right)\cdot 37^{2} + \left(3 a + 12\right)\cdot 37^{3} + \left(16 a + 1\right)\cdot 37^{4} + \left(32 a + 36\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 14 + \left(34 a + 14\right)\cdot 37 + \left(14 a + 23\right)\cdot 37^{2} + \left(13 a + 26\right)\cdot 37^{3} + \left(12 a + 13\right)\cdot 37^{4} + \left(10 a + 29\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 27 + \left(2 a + 19\right)\cdot 37 + \left(28 a + 25\right)\cdot 37^{2} + \left(33 a + 16\right)\cdot 37^{3} + \left(20 a + 25\right)\cdot 37^{4} + \left(4 a + 1\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 27\cdot 37 + 31\cdot 37^{2} + 34\cdot 37^{3} + 13\cdot 37^{4} + 12\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 29\cdot 37 + 30\cdot 37^{2} + 28\cdot 37^{3} + 6\cdot 37^{4} + 10\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 34 + \left(2 a + 34\right)\cdot 37 + \left(22 a + 11\right)\cdot 37^{2} + \left(23 a + 28\right)\cdot 37^{3} + \left(24 a + 12\right)\cdot 37^{4} + \left(26 a + 21\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,4,3,6,5,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
