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