Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 36\cdot 37 + 24\cdot 37^{2} + 19\cdot 37^{3} + 28\cdot 37^{4} + 30\cdot 37^{5} + 36\cdot 37^{6} + 30\cdot 37^{7} + 4\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 2 + \left(5 a + 2\right)\cdot 37 + \left(11 a + 36\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(30 a + 26\right)\cdot 37^{4} + \left(11 a + 28\right)\cdot 37^{5} + \left(12 a + 36\right)\cdot 37^{6} + \left(33 a + 31\right)\cdot 37^{7} + \left(16 a + 19\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 12\cdot 37^{2} + 17\cdot 37^{3} + 8\cdot 37^{4} + 6\cdot 37^{5} + 6\cdot 37^{7} + 32\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 27 + \left(11 a + 29\right)\cdot 37 + 30 a\cdot 37^{2} + \left(35 a + 36\right)\cdot 37^{3} + \left(15 a + 22\right)\cdot 37^{4} + \left(24 a + 14\right)\cdot 37^{5} + \left(18 a + 30\right)\cdot 37^{6} + \left(4 a + 18\right)\cdot 37^{7} + \left(6 a + 8\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 36 + \left(31 a + 34\right)\cdot 37 + 25 a\cdot 37^{2} + \left(8 a + 14\right)\cdot 37^{3} + \left(6 a + 10\right)\cdot 37^{4} + \left(25 a + 8\right)\cdot 37^{5} + 24 a\cdot 37^{6} + \left(3 a + 5\right)\cdot 37^{7} + \left(20 a + 17\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 11 + \left(25 a + 7\right)\cdot 37 + \left(6 a + 36\right)\cdot 37^{2} + a\cdot 37^{3} + \left(21 a + 14\right)\cdot 37^{4} + \left(12 a + 22\right)\cdot 37^{5} + \left(18 a + 6\right)\cdot 37^{6} + \left(32 a + 18\right)\cdot 37^{7} + \left(30 a + 28\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,6)$ |
| $(1,4,2)(3,6,5)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,3,4)$ | $-1$ |
| $6$ | $4$ | $(1,6,3,4)(2,5)$ | $1$ |
| $8$ | $6$ | $(1,6,5,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
