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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 2\cdot 37 + 8\cdot 37^{2} + 12\cdot 37^{3} + 31\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + \left(2 a + 6\right)\cdot 37 + 36 a\cdot 37^{2} + \left(32 a + 32\right)\cdot 37^{3} + \left(25 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 17 + \left(34 a + 22\right)\cdot 37 + 30\cdot 37^{2} + \left(4 a + 16\right)\cdot 37^{3} + \left(11 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 35 + \left(6 a + 34\right)\cdot 37 + 36 a\cdot 37^{2} + \left(3 a + 4\right)\cdot 37^{3} + 30 a\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a + 24 + \left(30 a + 36\right)\cdot 37 + 27\cdot 37^{2} + \left(33 a + 20\right)\cdot 37^{3} + \left(6 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 8\cdot 37 + 6\cdot 37^{2} + 25\cdot 37^{3} + 6\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
| $(1,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
