Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 32 + \left(11 a + 5\right)\cdot 37 + \left(34 a + 11\right)\cdot 37^{2} + \left(8 a + 36\right)\cdot 37^{3} + \left(8 a + 24\right)\cdot 37^{4} + \left(17 a + 6\right)\cdot 37^{5} + \left(12 a + 2\right)\cdot 37^{6} + \left(3 a + 18\right)\cdot 37^{7} + \left(30 a + 15\right)\cdot 37^{8} + \left(2 a + 9\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 34\cdot 37 + 25\cdot 37^{2} + 29\cdot 37^{3} + 3\cdot 37^{4} + 37^{5} + 16\cdot 37^{6} + 3\cdot 37^{7} + 7\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 34 + \left(13 a + 19\right)\cdot 37 + \left(9 a + 6\right)\cdot 37^{2} + \left(8 a + 25\right)\cdot 37^{3} + \left(27 a + 23\right)\cdot 37^{4} + \left(12 a + 6\right)\cdot 37^{5} + \left(34 a + 30\right)\cdot 37^{6} + \left(14 a + 5\right)\cdot 37^{7} + \left(2 a + 21\right)\cdot 37^{8} + \left(4 a + 11\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 5 + \left(25 a + 31\right)\cdot 37 + \left(2 a + 25\right)\cdot 37^{2} + 28 a\cdot 37^{3} + \left(28 a + 12\right)\cdot 37^{4} + \left(19 a + 30\right)\cdot 37^{5} + \left(24 a + 34\right)\cdot 37^{6} + \left(33 a + 18\right)\cdot 37^{7} + \left(6 a + 21\right)\cdot 37^{8} + \left(34 a + 27\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 2\cdot 37 + 11\cdot 37^{2} + 7\cdot 37^{3} + 33\cdot 37^{4} + 35\cdot 37^{5} + 20\cdot 37^{6} + 33\cdot 37^{7} + 36\cdot 37^{8} + 29\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 3 + \left(23 a + 17\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(9 a + 13\right)\cdot 37^{4} + \left(24 a + 30\right)\cdot 37^{5} + \left(2 a + 6\right)\cdot 37^{6} + \left(22 a + 31\right)\cdot 37^{7} + \left(34 a + 15\right)\cdot 37^{8} + \left(32 a + 25\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,5,3)(2,6,4)$ |
| $(2,6)(3,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)(2,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
