Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 5 + \left(30 a + 28\right)\cdot 37 + \left(a + 22\right)\cdot 37^{2} + \left(24 a + 30\right)\cdot 37^{3} + \left(34 a + 14\right)\cdot 37^{4} + \left(32 a + 30\right)\cdot 37^{5} + \left(30 a + 35\right)\cdot 37^{6} + \left(28 a + 15\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 6 + 28 a\cdot 37 + \left(21 a + 21\right)\cdot 37^{2} + \left(13 a + 7\right)\cdot 37^{3} + \left(3 a + 11\right)\cdot 37^{4} + \left(5 a + 32\right)\cdot 37^{5} + \left(35 a + 8\right)\cdot 37^{6} + \left(15 a + 32\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 9\cdot 37 + 29\cdot 37^{2} + 3\cdot 37^{3} + 13\cdot 37^{4} + 28\cdot 37^{5} + 8\cdot 37^{6} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 24 + \left(6 a + 24\right)\cdot 37 + \left(35 a + 36\right)\cdot 37^{2} + \left(12 a + 13\right)\cdot 37^{3} + \left(2 a + 18\right)\cdot 37^{4} + \left(4 a + 16\right)\cdot 37^{5} + \left(6 a + 15\right)\cdot 37^{6} + \left(8 a + 26\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 18\cdot 37 + 32\cdot 37^{2} + 14\cdot 37^{3} + 5\cdot 37^{4} + 28\cdot 37^{5} + 8\cdot 37^{6} + 12\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 5 + \left(8 a + 30\right)\cdot 37 + \left(15 a + 5\right)\cdot 37^{2} + \left(23 a + 3\right)\cdot 37^{3} + \left(33 a + 11\right)\cdot 37^{4} + \left(31 a + 12\right)\cdot 37^{5} + \left(a + 33\right)\cdot 37^{6} + \left(21 a + 23\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(2,6)$ |
| $(1,3,2)(4,5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$-3$ |
| $3$ |
$2$ |
$(3,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(3,5)$ |
$-1$ |
| $4$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,3,6,4,5,2)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,5,4,6,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
