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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 3 + \left(11 a + 21\right)\cdot 37 + \left(24 a + 25\right)\cdot 37^{2} + \left(19 a + 16\right)\cdot 37^{3} + \left(2 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 33 + \left(25 a + 4\right)\cdot 37 + 12 a\cdot 37^{2} + \left(17 a + 34\right)\cdot 37^{3} + \left(34 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 18 + \left(24 a + 13\right)\cdot 37 + 8 a\cdot 37^{2} + \left(25 a + 21\right)\cdot 37^{3} + \left(15 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 22 + \left(12 a + 34\right)\cdot 37 + \left(28 a + 10\right)\cdot 37^{2} + \left(11 a + 2\right)\cdot 37^{3} + \left(21 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
