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:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 9 + \left(15 a + 7\right)\cdot 37 + \left(20 a + 27\right)\cdot 37^{2} + \left(a + 8\right)\cdot 37^{3} + \left(18 a + 10\right)\cdot 37^{4} + \left(24 a + 25\right)\cdot 37^{5} + \left(7 a + 14\right)\cdot 37^{6} + \left(21 a + 12\right)\cdot 37^{7} + \left(10 a + 17\right)\cdot 37^{8} + \left(20 a + 29\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 3 + \left(21 a + 15\right)\cdot 37 + \left(16 a + 19\right)\cdot 37^{2} + \left(35 a + 31\right)\cdot 37^{3} + \left(18 a + 6\right)\cdot 37^{4} + \left(12 a + 31\right)\cdot 37^{5} + \left(29 a + 20\right)\cdot 37^{6} + \left(15 a + 15\right)\cdot 37^{7} + \left(26 a + 1\right)\cdot 37^{8} + \left(16 a + 26\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 6\cdot 37 + 21\cdot 37^{2} + 13\cdot 37^{3} + 19\cdot 37^{4} + 35\cdot 37^{5} + 13\cdot 37^{6} + 6\cdot 37^{7} + 13\cdot 37^{8} + 22\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 28 + \left(21 a + 29\right)\cdot 37 + \left(16 a + 9\right)\cdot 37^{2} + \left(35 a + 28\right)\cdot 37^{3} + \left(18 a + 26\right)\cdot 37^{4} + \left(12 a + 11\right)\cdot 37^{5} + \left(29 a + 22\right)\cdot 37^{6} + \left(15 a + 24\right)\cdot 37^{7} + \left(26 a + 19\right)\cdot 37^{8} + \left(16 a + 7\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 34 + \left(15 a + 21\right)\cdot 37 + \left(20 a + 17\right)\cdot 37^{2} + \left(a + 5\right)\cdot 37^{3} + \left(18 a + 30\right)\cdot 37^{4} + \left(24 a + 5\right)\cdot 37^{5} + \left(7 a + 16\right)\cdot 37^{6} + \left(21 a + 21\right)\cdot 37^{7} + \left(10 a + 35\right)\cdot 37^{8} + \left(20 a + 10\right)\cdot 37^{9} +O\left(37^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 30\cdot 37 + 15\cdot 37^{2} + 23\cdot 37^{3} + 17\cdot 37^{4} + 37^{5} + 23\cdot 37^{6} + 30\cdot 37^{7} + 23\cdot 37^{8} + 14\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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $1$ |
| $8$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
