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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 16\cdot 37 + 18\cdot 37^{2} + 14\cdot 37^{3} + 29\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 8\cdot 37 + 22\cdot 37^{2} + 11\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 15 + \left(5 a + 25\right)\cdot 37 + \left(26 a + 18\right)\cdot 37^{2} + \left(10 a + 13\right)\cdot 37^{3} + \left(26 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 1 + \left(9 a + 10\right)\cdot 37 + \left(27 a + 28\right)\cdot 37^{2} + \left(12 a + 8\right)\cdot 37^{3} + \left(36 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 13 + \left(31 a + 29\right)\cdot 37 + \left(10 a + 6\right)\cdot 37^{2} + \left(26 a + 30\right)\cdot 37^{3} + \left(10 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 35 + \left(27 a + 21\right)\cdot 37 + \left(9 a + 16\right)\cdot 37^{2} + \left(24 a + 32\right)\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,5)$ |
$2$ |
| $9$ |
$2$ |
$(3,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,6,5,2)$ |
$0$ |
| $12$ |
$6$ |
$(2,4,6)(3,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
