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:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 27 + \left(31 a + 24\right)\cdot 37 + \left(19 a + 17\right)\cdot 37^{2} + \left(21 a + 2\right)\cdot 37^{3} + \left(21 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 17 + \left(4 a + 25\right)\cdot 37 + \left(30 a + 20\right)\cdot 37^{2} + \left(23 a + 31\right)\cdot 37^{3} + \left(28 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 20\cdot 37 + 27\cdot 37^{2} + 2\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 24\cdot 37 + 27\cdot 37^{2} + 19\cdot 37^{3} + 33\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 30 + \left(5 a + 28\right)\cdot 37 + \left(17 a + 28\right)\cdot 37^{2} + \left(15 a + 31\right)\cdot 37^{3} + \left(15 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 23 + \left(32 a + 23\right)\cdot 37 + \left(6 a + 25\right)\cdot 37^{2} + \left(13 a + 22\right)\cdot 37^{3} + \left(8 a + 28\right)\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 value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,5)$ | $0$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
