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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 3\cdot 37 + 6\cdot 37^{2} + 10\cdot 37^{3} + 12\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 23 + \left(34 a + 7\right)\cdot 37 + \left(10 a + 12\right)\cdot 37^{2} + \left(8 a + 21\right)\cdot 37^{3} + \left(29 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 15 + \left(2 a + 1\right)\cdot 37 + \left(26 a + 21\right)\cdot 37^{2} + \left(28 a + 6\right)\cdot 37^{3} + \left(7 a + 21\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 19\cdot 37 + 4\cdot 37^{2} + 8\cdot 37^{3} + 3\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 29 + \left(16 a + 10\right)\cdot 37 + \left(31 a + 34\right)\cdot 37^{2} + \left(27 a + 10\right)\cdot 37^{3} + \left(5 a + 9\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 20 + \left(20 a + 31\right)\cdot 37 + \left(5 a + 32\right)\cdot 37^{2} + \left(9 a + 16\right)\cdot 37^{3} + \left(31 a + 4\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,6,5)$ |
| $(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $20$ | $3$ | $(1,2,4)(3,6,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,5,2,4)$ | $0$ |
| $12$ | $5$ | $(1,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
