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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 15 + 35\cdot 37 + \left(18 a + 13\right)\cdot 37^{2} + \left(7 a + 24\right)\cdot 37^{3} + \left(28 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 2 + \left(9 a + 9\right)\cdot 37 + \left(33 a + 29\right)\cdot 37^{2} + \left(14 a + 14\right)\cdot 37^{3} + \left(15 a + 28\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 30 + \left(36 a + 23\right)\cdot 37 + \left(18 a + 11\right)\cdot 37^{2} + \left(29 a + 36\right)\cdot 37^{3} + \left(8 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 19\cdot 37 + 14\cdot 37^{2} + 31\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 24 + \left(27 a + 23\right)\cdot 37 + \left(3 a + 4\right)\cdot 37^{2} + \left(22 a + 4\right)\cdot 37^{3} + \left(21 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
