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:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 21 + \left(2 a + 8\right)\cdot 37 + \left(3 a + 11\right)\cdot 37^{2} + \left(27 a + 23\right)\cdot 37^{3} + \left(35 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 22 + \left(34 a + 28\right)\cdot 37 + \left(33 a + 20\right)\cdot 37^{2} + \left(9 a + 17\right)\cdot 37^{3} + \left(a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 27 + \left(27 a + 21\right)\cdot 37 + \left(34 a + 25\right)\cdot 37^{2} + \left(16 a + 4\right)\cdot 37^{3} + \left(7 a + 9\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 27\cdot 37^{2} + 27\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 10 + \left(9 a + 14\right)\cdot 37 + \left(2 a + 26\right)\cdot 37^{2} + 20 a\cdot 37^{3} + \left(29 a + 22\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.
