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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 34\cdot 37 + 8\cdot 37^{2} + 8\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 10 + 2 a\cdot 37 + \left(a + 8\right)\cdot 37^{2} + \left(9 a + 9\right)\cdot 37^{3} + \left(21 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 10 + \left(14 a + 23\right)\cdot 37 + \left(4 a + 13\right)\cdot 37^{2} + \left(11 a + 27\right)\cdot 37^{3} + \left(30 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 13 + \left(34 a + 36\right)\cdot 37 + \left(35 a + 9\right)\cdot 37^{2} + \left(27 a + 7\right)\cdot 37^{3} + \left(15 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 35 + \left(22 a + 11\right)\cdot 37 + \left(32 a + 16\right)\cdot 37^{2} + \left(25 a + 30\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 + 5\cdot 37 + 17\cdot 37^{2} + 28\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6,3)$ |
| $(1,4,5)(2,3,6)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,6)(3,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,6,3)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
