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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 4 + \left(30 a + 13\right)\cdot 37 + \left(10 a + 29\right)\cdot 37^{2} + \left(28 a + 16\right)\cdot 37^{3} + \left(21 a + 12\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 22 + \left(30 a + 8\right)\cdot 37 + \left(10 a + 26\right)\cdot 37^{2} + \left(28 a + 9\right)\cdot 37^{3} + \left(21 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 2 + \left(6 a + 7\right)\cdot 37 + \left(26 a + 12\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(15 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 25 + \left(6 a + 29\right)\cdot 37 + \left(26 a + 4\right)\cdot 37^{2} + \left(8 a + 8\right)\cdot 37^{3} + \left(15 a + 34\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 18 + \left(30 a + 27\right)\cdot 37 + \left(10 a + 36\right)\cdot 37^{2} + \left(28 a + 4\right)\cdot 37^{3} + \left(21 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 6 + \left(6 a + 25\right)\cdot 37 + \left(26 a + 1\right)\cdot 37^{2} + \left(8 a + 1\right)\cdot 37^{3} + \left(15 a + 21\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2,4,5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,2,4,5,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,5,4,2,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
