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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 9 + \left(26 a + 1\right)\cdot 37 + \left(19 a + 33\right)\cdot 37^{2} + \left(22 a + 27\right)\cdot 37^{3} + \left(6 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 31 + \left(10 a + 9\right)\cdot 37 + \left(17 a + 11\right)\cdot 37^{2} + \left(14 a + 24\right)\cdot 37^{3} + \left(30 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 19 + \left(24 a + 23\right)\cdot 37 + 15\cdot 37^{2} + \left(23 a + 3\right)\cdot 37^{3} + \left(13 a + 35\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 31\cdot 37 + 33\cdot 37^{2} + 14\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 28\cdot 37 + 23\cdot 37^{2} + 19\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 36 + \left(12 a + 16\right)\cdot 37 + \left(36 a + 30\right)\cdot 37^{2} + \left(13 a + 20\right)\cdot 37^{3} + \left(23 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,6,2,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $15$ | $2$ | $(1,5)(2,6)$ | $1$ |
| $20$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $30$ | $4$ | $(1,6,5,2)$ | $1$ |
| $24$ | $5$ | $(1,2,5,3,4)$ | $0$ |
| $20$ | $6$ | $(1,6,2,4,3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
