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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 21 + \left(5 a + 28\right)\cdot 37 + \left(2 a + 12\right)\cdot 37^{2} + \left(6 a + 24\right)\cdot 37^{3} + \left(5 a + 4\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 35\cdot 37 + 34\cdot 37^{2} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 8 + \left(26 a + 29\right)\cdot 37 + \left(24 a + 16\right)\cdot 37^{2} + \left(14 a + 27\right)\cdot 37^{3} + \left(8 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 15 + \left(10 a + 12\right)\cdot 37 + \left(12 a + 15\right)\cdot 37^{2} + \left(22 a + 24\right)\cdot 37^{3} + \left(28 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 19\cdot 37 + 15\cdot 37^{2} + 24\cdot 37^{3} + 9\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 26 + \left(31 a + 22\right)\cdot 37 + \left(34 a + 15\right)\cdot 37^{2} + \left(30 a + 9\right)\cdot 37^{3} + \left(31 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $0$ |
| $2$ | $3$ | $(1,5,6)(2,3,4)$ | $-1$ |
| $2$ | $6$ | $(1,3,5,4,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
