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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 3 + \left(2 a + 17\right)\cdot 37 + \left(7 a + 27\right)\cdot 37^{2} + \left(14 a + 18\right)\cdot 37^{3} + \left(29 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 16 + \left(34 a + 34\right)\cdot 37 + \left(29 a + 15\right)\cdot 37^{2} + \left(22 a + 31\right)\cdot 37^{3} + \left(7 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 6\cdot 37 + 29\cdot 37^{2} + 14\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 16 + \left(21 a + 13\right)\cdot 37 + \left(6 a + 1\right)\cdot 37^{2} + 7 a\cdot 37^{3} + \left(18 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 7 + \left(15 a + 17\right)\cdot 37 + \left(30 a + 6\right)\cdot 37^{2} + \left(29 a + 22\right)\cdot 37^{3} + \left(18 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 22\cdot 37 + 30\cdot 37^{2} + 23\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(4,5)$ |
$-2$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(3,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,6)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
