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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 30 + \left(15 a + 16\right)\cdot 37 + \left(17 a + 28\right)\cdot 37^{2} + \left(32 a + 17\right)\cdot 37^{3} + \left(26 a + 36\right)\cdot 37^{4} + \left(25 a + 35\right)\cdot 37^{5} + \left(9 a + 11\right)\cdot 37^{6} + \left(27 a + 24\right)\cdot 37^{7} + \left(6 a + 18\right)\cdot 37^{8} + \left(31 a + 33\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 6\cdot 37 + 3\cdot 37^{2} + 36\cdot 37^{3} + 24\cdot 37^{4} + 4\cdot 37^{5} + 4\cdot 37^{6} + 24\cdot 37^{7} + 16\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 18 + 6 a\cdot 37 + \left(6 a + 28\right)\cdot 37^{2} + \left(30 a + 16\right)\cdot 37^{3} + \left(7 a + 36\right)\cdot 37^{4} + \left(3 a + 15\right)\cdot 37^{5} + \left(31 a + 13\right)\cdot 37^{6} + \left(30 a + 9\right)\cdot 37^{7} + \left(10 a + 12\right)\cdot 37^{8} + \left(16 a + 28\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 7 + \left(21 a + 20\right)\cdot 37 + \left(19 a + 8\right)\cdot 37^{2} + \left(4 a + 19\right)\cdot 37^{3} + 10 a\cdot 37^{4} + \left(11 a + 1\right)\cdot 37^{5} + \left(27 a + 25\right)\cdot 37^{6} + \left(9 a + 12\right)\cdot 37^{7} + \left(30 a + 18\right)\cdot 37^{8} + \left(5 a + 3\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 30\cdot 37 + 33\cdot 37^{2} + 12\cdot 37^{4} + 32\cdot 37^{5} + 32\cdot 37^{6} + 12\cdot 37^{7} + 36\cdot 37^{8} + 20\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 19 + \left(30 a + 36\right)\cdot 37 + \left(30 a + 8\right)\cdot 37^{2} + \left(6 a + 20\right)\cdot 37^{3} + 29 a\cdot 37^{4} + \left(33 a + 21\right)\cdot 37^{5} + \left(5 a + 23\right)\cdot 37^{6} + \left(6 a + 27\right)\cdot 37^{7} + \left(26 a + 24\right)\cdot 37^{8} + \left(20 a + 8\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $1$ |
| $8$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
