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:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 6 + \left(35 a + 9\right)\cdot 37 + \left(23 a + 15\right)\cdot 37^{2} + \left(20 a + 11\right)\cdot 37^{3} + a\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 15 + \left(5 a + 1\right)\cdot 37 + \left(27 a + 22\right)\cdot 37^{2} + \left(17 a + 24\right)\cdot 37^{3} + \left(35 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 35 + \left(a + 6\right)\cdot 37 + \left(13 a + 1\right)\cdot 37^{2} + \left(16 a + 33\right)\cdot 37^{3} + \left(35 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 29 + \left(31 a + 1\right)\cdot 37 + \left(9 a + 14\right)\cdot 37^{2} + \left(19 a + 31\right)\cdot 37^{3} + \left(a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 17\cdot 37 + 21\cdot 37^{2} + 10\cdot 37^{3} + 11\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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
