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:
| $r_{ 1 }$ |
$=$ |
$ a + 3 + \left(34 a + 36\right)\cdot 37 + \left(27 a + 26\right)\cdot 37^{2} + 4 a\cdot 37^{3} + \left(27 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 17\cdot 37 + 9\cdot 37^{2} + 23\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 a + 7 + \left(2 a + 23\right)\cdot 37 + \left(9 a + 30\right)\cdot 37^{2} + \left(32 a + 28\right)\cdot 37^{3} + \left(9 a + 17\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 19 + \left(14 a + 33\right)\cdot 37 + \left(33 a + 9\right)\cdot 37^{2} + \left(35 a + 8\right)\cdot 37^{3} + \left(25 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 12\cdot 37 + 16\cdot 37^{2} + 5\cdot 37^{3} + 29\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 32 + \left(22 a + 24\right)\cdot 37 + \left(3 a + 17\right)\cdot 37^{2} + \left(a + 7\right)\cdot 37^{3} + \left(11 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(2,3)(5,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,4,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
