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