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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 9 + \left(21 a + 32\right)\cdot 37 + \left(4 a + 18\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + \left(35 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 31 + \left(15 a + 20\right)\cdot 37 + \left(32 a + 15\right)\cdot 37^{2} + \left(5 a + 17\right)\cdot 37^{3} + \left(a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 18 + \left(35 a + 13\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(12 a + 29\right)\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 1 + \left(16 a + 12\right)\cdot 37 + \left(12 a + 27\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(24 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 23 + \left(a + 16\right)\cdot 37 + \left(9 a + 32\right)\cdot 37^{2} + \left(24 a + 15\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(24 a + 23\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(12 a + 35\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,3)$ |
| $(1,3,6,2,5,4)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,6,5)(2,4,3)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,5,6)(2,3,4)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,5,6)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(1,6,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,5,6)(2,4,3)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,3,6,2,5,4)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,4,5,2,6,3)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
