Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 20 + \left(36 a + 32\right)\cdot 37 + \left(5 a + 9\right)\cdot 37^{2} + \left(29 a + 21\right)\cdot 37^{3} + \left(30 a + 13\right)\cdot 37^{4} + \left(5 a + 7\right)\cdot 37^{5} + \left(33 a + 28\right)\cdot 37^{6} + \left(18 a + 32\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 14\cdot 37 + 28\cdot 37^{2} + 12\cdot 37^{3} + 19\cdot 37^{4} + 11\cdot 37^{5} + 27\cdot 37^{6} + 8\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 14 + \left(36 a + 35\right)\cdot 37 + \left(5 a + 22\right)\cdot 37^{2} + \left(29 a + 21\right)\cdot 37^{3} + \left(30 a + 18\right)\cdot 37^{4} + \left(5 a + 19\right)\cdot 37^{5} + \left(33 a + 23\right)\cdot 37^{6} + \left(18 a + 16\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 5 + 25\cdot 37 + \left(31 a + 10\right)\cdot 37^{2} + \left(7 a + 21\right)\cdot 37^{3} + \left(6 a + 1\right)\cdot 37^{4} + \left(31 a + 12\right)\cdot 37^{5} + \left(3 a + 2\right)\cdot 37^{6} + \left(18 a + 22\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 11 + 22\cdot 37 + \left(31 a + 34\right)\cdot 37^{2} + \left(7 a + 20\right)\cdot 37^{3} + \left(6 a + 33\right)\cdot 37^{4} + \left(31 a + 36\right)\cdot 37^{5} + \left(3 a + 6\right)\cdot 37^{6} + \left(18 a + 1\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 17\cdot 37 + 4\cdot 37^{2} + 13\cdot 37^{3} + 24\cdot 37^{4} + 23\cdot 37^{5} + 22\cdot 37^{6} + 29\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,2)(3,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
