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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 12 + \left(32 a + 13\right)\cdot 37 + \left(8 a + 13\right)\cdot 37^{2} + \left(28 a + 30\right)\cdot 37^{3} + \left(7 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 10 + \left(30 a + 3\right)\cdot 37 + \left(14 a + 35\right)\cdot 37^{2} + \left(10 a + 4\right)\cdot 37^{3} + \left(3 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 9\cdot 37 + 22\cdot 37^{2} + 10\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 16\cdot 37 + 33\cdot 37^{2} + 9\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 13 + \left(6 a + 3\right)\cdot 37 + \left(22 a + 27\right)\cdot 37^{2} + \left(26 a + 31\right)\cdot 37^{3} + \left(33 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 20 + \left(4 a + 28\right)\cdot 37 + \left(28 a + 16\right)\cdot 37^{2} + \left(8 a + 23\right)\cdot 37^{3} + \left(29 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,5)$ |
$0$ |
| $9$ |
$2$ |
$(3,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,4,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,4,5,6,2)$ |
$1$ |
| $12$ |
$6$ |
$(1,4,6)(3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
