Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 a + 6 + \left(22 a + 4\right)\cdot 37 + \left(22 a + 26\right)\cdot 37^{2} + \left(5 a + 26\right)\cdot 37^{3} + \left(16 a + 24\right)\cdot 37^{4} + \left(30 a + 1\right)\cdot 37^{5} + \left(16 a + 5\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 3 + \left(8 a + 34\right)\cdot 37 + \left(29 a + 3\right)\cdot 37^{2} + \left(32 a + 25\right)\cdot 37^{3} + 8\cdot 37^{4} + \left(8 a + 20\right)\cdot 37^{5} + \left(11 a + 27\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 24\cdot 37 + 27\cdot 37^{2} + 20\cdot 37^{3} + 2\cdot 37^{4} + 2\cdot 37^{5} + 27\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 13 + \left(14 a + 8\right)\cdot 37 + \left(14 a + 20\right)\cdot 37^{2} + \left(31 a + 26\right)\cdot 37^{3} + \left(20 a + 9\right)\cdot 37^{4} + \left(6 a + 33\right)\cdot 37^{5} + \left(20 a + 4\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 23 + \left(28 a + 24\right)\cdot 37 + \left(7 a + 1\right)\cdot 37^{2} + \left(4 a + 16\right)\cdot 37^{3} + \left(36 a + 16\right)\cdot 37^{4} + \left(28 a + 14\right)\cdot 37^{5} + \left(25 a + 27\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 + 15\cdot 37 + 31\cdot 37^{2} + 32\cdot 37^{3} + 11\cdot 37^{4} + 2\cdot 37^{5} + 19\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(5,6)$ |
| $(2,5)(3,4)$ |
| $(1,5,3,2,4,6)$ |
| $(2,6)(3,4)$ |
| $(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $9$ |
$2$ |
$(2,5)(3,4)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$-2$ |
| $2$ |
$3$ |
$(1,4,3)(2,6,5)$ |
$-2$ |
| $4$ |
$3$ |
$(2,6,5)$ |
$1$ |
| $6$ |
$6$ |
$(1,5,3,2,4,6)$ |
$0$ |
| $6$ |
$6$ |
$(1,5,4,2,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
