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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 23 + \left(18 a + 28\right)\cdot 37 + \left(33 a + 14\right)\cdot 37^{2} + \left(15 a + 34\right)\cdot 37^{3} + \left(21 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 23 + \left(18 a + 9\right)\cdot 37 + \left(3 a + 16\right)\cdot 37^{2} + \left(21 a + 20\right)\cdot 37^{3} + \left(15 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + \left(18 a + 6\right)\cdot 37 + \left(33 a + 22\right)\cdot 37^{2} + \left(15 a + 22\right)\cdot 37^{3} + \left(21 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 5 + \left(18 a + 14\right)\cdot 37 + \left(3 a + 19\right)\cdot 37^{2} + \left(21 a + 27\right)\cdot 37^{3} + \left(15 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 4 + \left(18 a + 24\right)\cdot 37 + \left(33 a + 11\right)\cdot 37^{2} + \left(15 a + 27\right)\cdot 37^{3} + \left(21 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 19 + \left(18 a + 28\right)\cdot 37 + \left(3 a + 26\right)\cdot 37^{2} + \left(21 a + 15\right)\cdot 37^{3} + \left(15 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,3,4,5,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,5,4,3,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
