Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 8\cdot 37 + 21\cdot 37^{2} + 8\cdot 37^{3} + 3\cdot 37^{4} + 20\cdot 37^{5} + 7\cdot 37^{6} + 37^{7} + 12\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 28\cdot 37 + 15\cdot 37^{2} + 28\cdot 37^{3} + 33\cdot 37^{4} + 16\cdot 37^{5} + 29\cdot 37^{6} + 35\cdot 37^{7} + 24\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 9 + \left(12 a + 15\right)\cdot 37 + \left(23 a + 33\right)\cdot 37^{2} + \left(21 a + 23\right)\cdot 37^{3} + \left(27 a + 29\right)\cdot 37^{4} + \left(25 a + 17\right)\cdot 37^{5} + \left(34 a + 17\right)\cdot 37^{6} + \left(8 a + 36\right)\cdot 37^{7} + \left(19 a + 2\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 27 + \left(a + 12\right)\cdot 37 + \left(33 a + 27\right)\cdot 37^{2} + \left(13 a + 25\right)\cdot 37^{3} + \left(32 a + 34\right)\cdot 37^{4} + \left(2 a + 28\right)\cdot 37^{5} + \left(36 a + 21\right)\cdot 37^{6} + \left(18 a + 35\right)\cdot 37^{7} + \left(27 a + 9\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 11 + \left(35 a + 24\right)\cdot 37 + \left(3 a + 9\right)\cdot 37^{2} + \left(23 a + 11\right)\cdot 37^{3} + \left(4 a + 2\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} + 15\cdot 37^{6} + \left(18 a + 1\right)\cdot 37^{7} + \left(9 a + 27\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 29 + \left(24 a + 21\right)\cdot 37 + \left(13 a + 3\right)\cdot 37^{2} + \left(15 a + 13\right)\cdot 37^{3} + \left(9 a + 7\right)\cdot 37^{4} + \left(11 a + 19\right)\cdot 37^{5} + \left(2 a + 19\right)\cdot 37^{6} + 28 a\cdot 37^{7} + \left(17 a + 34\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5)$ |
| $(1,2)$ |
| $(3,6)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $4$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,5,6,2,4,3)$ |
$0$ |
| $4$ |
$6$ |
$(1,3,4,2,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
