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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 17\cdot 37 + 12\cdot 37^{2} + 32\cdot 37^{3} + 37^{4} + 9\cdot 37^{5} + 4\cdot 37^{6} + 11\cdot 37^{7} + 4\cdot 37^{8} + 32\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 13 + \left(6 a + 27\right)\cdot 37 + \left(35 a + 29\right)\cdot 37^{2} + \left(20 a + 23\right)\cdot 37^{3} + \left(23 a + 15\right)\cdot 37^{4} + \left(28 a + 24\right)\cdot 37^{5} + 26 a\cdot 37^{6} + \left(27 a + 29\right)\cdot 37^{7} + \left(4 a + 20\right)\cdot 37^{8} + \left(34 a + 16\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 10 + \left(6 a + 5\right)\cdot 37 + \left(35 a + 24\right)\cdot 37^{2} + \left(20 a + 25\right)\cdot 37^{3} + \left(23 a + 24\right)\cdot 37^{4} + \left(28 a + 22\right)\cdot 37^{5} + \left(26 a + 17\right)\cdot 37^{6} + \left(27 a + 18\right)\cdot 37^{7} + \left(4 a + 20\right)\cdot 37^{8} + \left(34 a + 29\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 2\cdot 37 + 18\cdot 37^{2} + 30\cdot 37^{3} + 29\cdot 37^{4} + 10\cdot 37^{5} + 24\cdot 37^{6} + 21\cdot 37^{7} + 4\cdot 37^{8} + 19\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 21 + \left(30 a + 18\right)\cdot 37 + \left(a + 10\right)\cdot 37^{2} + 16 a\cdot 37^{3} + \left(13 a + 24\right)\cdot 37^{4} + \left(8 a + 2\right)\cdot 37^{5} + \left(10 a + 22\right)\cdot 37^{6} + \left(9 a + 28\right)\cdot 37^{7} + \left(32 a + 11\right)\cdot 37^{8} + \left(2 a + 13\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 24 + \left(30 a + 3\right)\cdot 37 + \left(a + 16\right)\cdot 37^{2} + \left(16 a + 35\right)\cdot 37^{3} + \left(13 a + 14\right)\cdot 37^{4} + \left(8 a + 4\right)\cdot 37^{5} + \left(10 a + 5\right)\cdot 37^{6} + \left(9 a + 2\right)\cdot 37^{7} + \left(32 a + 12\right)\cdot 37^{8} + 2 a\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
