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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 10 + \left(35 a + 19\right)\cdot 37 + \left(23 a + 4\right)\cdot 37^{2} + \left(12 a + 19\right)\cdot 37^{3} + \left(25 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 18 + \left(29 a + 1\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(25 a + 13\right)\cdot 37^{3} + \left(33 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 27 + \left(7 a + 16\right)\cdot 37 + \left(30 a + 3\right)\cdot 37^{2} + \left(11 a + 33\right)\cdot 37^{3} + \left(3 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 34 + \left(a + 5\right)\cdot 37 + \left(13 a + 28\right)\cdot 37^{2} + \left(24 a + 8\right)\cdot 37^{3} + 11 a\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 20 + \left(21 a + 12\right)\cdot 37 + \left(3 a + 19\right)\cdot 37^{2} + \left(2 a + 34\right)\cdot 37^{3} + 8\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 3 + \left(15 a + 18\right)\cdot 37 + \left(33 a + 12\right)\cdot 37^{2} + \left(34 a + 2\right)\cdot 37^{3} + \left(36 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,5,4)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,6,2)(3,4,5)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,3,6,4,2,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,2,4,6,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
