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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 10\cdot 37 + 12\cdot 37^{2} + 9\cdot 37^{3} + 11\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 11 + \left(13 a + 19\right)\cdot 37 + \left(2 a + 24\right)\cdot 37^{2} + \left(5 a + 29\right)\cdot 37^{3} + \left(34 a + 17\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 18 + \left(23 a + 24\right)\cdot 37 + \left(34 a + 20\right)\cdot 37^{2} + \left(31 a + 10\right)\cdot 37^{3} + \left(2 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 26 + \left(6 a + 11\right)\cdot 37 + \left(26 a + 4\right)\cdot 37^{2} + \left(15 a + 32\right)\cdot 37^{3} + \left(15 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 35\cdot 37 + 20\cdot 37^{2} + 34\cdot 37^{3} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 31 + \left(30 a + 9\right)\cdot 37 + \left(10 a + 28\right)\cdot 37^{2} + \left(21 a + 31\right)\cdot 37^{3} + \left(21 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$0$ |
| $6$ |
$2$ |
$(1,2)$ |
$-2$ |
| $9$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,5,2,4)(3,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,2,6,3,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,2)(4,5,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
