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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 35\cdot 37 + 36\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 5 + \left(23 a + 33\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(6 a + 24\right)\cdot 37^{3} + \left(a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 11\cdot 37 + 34\cdot 37^{2} + 4\cdot 37^{3} + 11\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 6 + \left(12 a + 5\right)\cdot 37 + \left(26 a + 20\right)\cdot 37^{2} + \left(14 a + 22\right)\cdot 37^{3} + \left(8 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 18 + \left(13 a + 23\right)\cdot 37 + \left(30 a + 9\right)\cdot 37^{2} + \left(30 a + 5\right)\cdot 37^{3} + \left(35 a + 4\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 29 + \left(24 a + 2\right)\cdot 37 + \left(10 a + 2\right)\cdot 37^{2} + \left(22 a + 18\right)\cdot 37^{3} + \left(28 a + 14\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)(3,4)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,6,2,3,5,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
