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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 37 + 14\cdot 37^{2} + 37^{3} + 29\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 28 + \left(21 a + 1\right)\cdot 37 + \left(22 a + 5\right)\cdot 37^{2} + \left(18 a + 33\right)\cdot 37^{3} + \left(28 a + 3\right)\cdot 37^{4} + 33 a\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 22 + \left(15 a + 33\right)\cdot 37 + \left(14 a + 36\right)\cdot 37^{2} + \left(18 a + 10\right)\cdot 37^{3} + \left(8 a + 25\right)\cdot 37^{4} + \left(3 a + 32\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 32 + \left(15 a + 31\right)\cdot 37 + \left(14 a + 34\right)\cdot 37^{2} + \left(18 a + 30\right)\cdot 37^{3} + \left(8 a + 19\right)\cdot 37^{4} + \left(3 a + 11\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 22\cdot 37 + 16\cdot 37^{2} + 18\cdot 37^{3} + 34\cdot 37^{4} + 35\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 1 + 21 a\cdot 37 + \left(22 a + 3\right)\cdot 37^{2} + \left(18 a + 16\right)\cdot 37^{3} + \left(28 a + 35\right)\cdot 37^{4} + \left(33 a + 15\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,6)$ |
| $(1,2,4,5,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(2,3)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,6)(2,5,3)$ |
$-1$ |
| $2$ |
$6$ |
$(1,2,4,5,6,3)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
