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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 23\cdot 37 + 30\cdot 37^{2} + 6\cdot 37^{3} + 31\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 27\cdot 37 + 15\cdot 37^{2} + 36\cdot 37^{3} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + \left(8 a + 36\right)\cdot 37 + \left(11 a + 21\right)\cdot 37^{2} + \left(18 a + 2\right)\cdot 37^{3} + \left(5 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 35 + \left(28 a + 14\right)\cdot 37 + \left(25 a + 21\right)\cdot 37^{2} + \left(18 a + 27\right)\cdot 37^{3} + \left(31 a + 4\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 23 + \left(5 a + 25\right)\cdot 37 + \left(29 a + 10\right)\cdot 37^{2} + \left(28 a + 31\right)\cdot 37^{3} + \left(14 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 24 + \left(31 a + 20\right)\cdot 37 + \left(7 a + 10\right)\cdot 37^{2} + \left(8 a + 6\right)\cdot 37^{3} + \left(22 a + 17\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $-2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
