Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 29 + \left(28 a + 4\right)\cdot 37 + \left(21 a + 8\right)\cdot 37^{2} + \left(6 a + 34\right)\cdot 37^{3} + \left(2 a + 27\right)\cdot 37^{4} + \left(5 a + 13\right)\cdot 37^{5} + 11\cdot 37^{6} + \left(17 a + 5\right)\cdot 37^{7} + \left(7 a + 20\right)\cdot 37^{8} + \left(14 a + 23\right)\cdot 37^{9} + \left(21 a + 15\right)\cdot 37^{10} + \left(17 a + 35\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 31 + \left(8 a + 25\right)\cdot 37 + \left(15 a + 29\right)\cdot 37^{2} + \left(30 a + 1\right)\cdot 37^{3} + \left(34 a + 30\right)\cdot 37^{4} + \left(31 a + 31\right)\cdot 37^{5} + \left(36 a + 6\right)\cdot 37^{6} + \left(19 a + 36\right)\cdot 37^{7} + \left(29 a + 32\right)\cdot 37^{8} + \left(22 a + 35\right)\cdot 37^{9} + \left(15 a + 12\right)\cdot 37^{10} + \left(19 a + 10\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 3\cdot 37 + 5\cdot 37^{2} + 29\cdot 37^{3} + 5\cdot 37^{5} + 14\cdot 37^{6} + 36\cdot 37^{7} + 25\cdot 37^{8} + 18\cdot 37^{9} + 15\cdot 37^{10} + 33\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 8 + \left(8 a + 32\right)\cdot 37 + \left(15 a + 28\right)\cdot 37^{2} + \left(30 a + 2\right)\cdot 37^{3} + \left(34 a + 9\right)\cdot 37^{4} + \left(31 a + 23\right)\cdot 37^{5} + \left(36 a + 25\right)\cdot 37^{6} + \left(19 a + 31\right)\cdot 37^{7} + \left(29 a + 16\right)\cdot 37^{8} + \left(22 a + 13\right)\cdot 37^{9} + \left(15 a + 21\right)\cdot 37^{10} + \left(19 a + 1\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 6 + \left(28 a + 11\right)\cdot 37 + \left(21 a + 7\right)\cdot 37^{2} + \left(6 a + 35\right)\cdot 37^{3} + \left(2 a + 6\right)\cdot 37^{4} + \left(5 a + 5\right)\cdot 37^{5} + 30\cdot 37^{6} + 17 a\cdot 37^{7} + \left(7 a + 4\right)\cdot 37^{8} + \left(14 a + 1\right)\cdot 37^{9} + \left(21 a + 24\right)\cdot 37^{10} + \left(17 a + 26\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 33\cdot 37 + 31\cdot 37^{2} + 7\cdot 37^{3} + 36\cdot 37^{4} + 31\cdot 37^{5} + 22\cdot 37^{6} + 11\cdot 37^{8} + 18\cdot 37^{9} + 21\cdot 37^{10} + 3\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(2,3)(5,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,4,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,4,3)(2,5)$ |
$-1$ |
| $8$ |
$6$ |
$(1,5,6,4,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
