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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 19\cdot 37 + 8\cdot 37^{2} + 27\cdot 37^{3} + 20\cdot 37^{4} + 23\cdot 37^{5} + 8\cdot 37^{6} + 13\cdot 37^{7} + 31\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 23 + \left(36 a + 20\right)\cdot 37 + \left(3 a + 12\right)\cdot 37^{2} + \left(34 a + 16\right)\cdot 37^{3} + \left(10 a + 27\right)\cdot 37^{4} + \left(29 a + 14\right)\cdot 37^{5} + \left(12 a + 23\right)\cdot 37^{6} + \left(26 a + 15\right)\cdot 37^{7} + \left(16 a + 27\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 20 + 29\cdot 37 + \left(33 a + 28\right)\cdot 37^{2} + 2 a\cdot 37^{3} + 26 a\cdot 37^{4} + \left(7 a + 10\right)\cdot 37^{5} + \left(24 a + 8\right)\cdot 37^{6} + \left(10 a + 34\right)\cdot 37^{7} + \left(20 a + 30\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 8 + \left(35 a + 18\right)\cdot 37 + \left(36 a + 33\right)\cdot 37^{2} + \left(17 a + 12\right)\cdot 37^{3} + \left(20 a + 18\right)\cdot 37^{4} + \left(19 a + 18\right)\cdot 37^{5} + \left(a + 21\right)\cdot 37^{6} + \left(21 a + 14\right)\cdot 37^{7} + \left(18 a + 2\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 4 + \left(a + 15\right)\cdot 37 + 34\cdot 37^{2} + \left(19 a + 10\right)\cdot 37^{3} + \left(16 a + 8\right)\cdot 37^{4} + \left(17 a + 2\right)\cdot 37^{5} + \left(35 a + 8\right)\cdot 37^{6} + \left(15 a + 23\right)\cdot 37^{7} + \left(18 a + 18\right)\cdot 37^{8} +O\left(37^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 8\cdot 37 + 30\cdot 37^{2} + 5\cdot 37^{3} + 36\cdot 37^{4} + 4\cdot 37^{5} + 4\cdot 37^{6} + 10\cdot 37^{7} +O\left(37^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,3,5,4)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,3,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,3,5,6,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
