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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 4\cdot 37 + 14\cdot 37^{2} + 28\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 15 + \left(14 a + 23\right)\cdot 37 + \left(4 a + 24\right)\cdot 37^{2} + \left(33 a + 31\right)\cdot 37^{3} + \left(15 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 25 + \left(6 a + 16\right)\cdot 37 + \left(21 a + 3\right)\cdot 37^{2} + \left(30 a + 14\right)\cdot 37^{3} + \left(25 a + 14\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 28 + \left(22 a + 14\right)\cdot 37 + \left(32 a + 27\right)\cdot 37^{2} + \left(3 a + 11\right)\cdot 37^{3} + \left(21 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 22\cdot 37 + 33\cdot 37^{2} + 20\cdot 37^{3} + 7\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 36 + \left(30 a + 29\right)\cdot 37 + \left(15 a + 7\right)\cdot 37^{2} + \left(6 a + 4\right)\cdot 37^{3} + \left(11 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)$ |
| $(1,4,5,6,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $15$ | $2$ | $(1,5)(2,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ |
| $30$ | $4$ | $(1,6,4,5)$ | $0$ |
| $24$ | $5$ | $(1,3,2,4,6)$ | $1$ |
| $20$ | $6$ | $(1,3,2,6,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
