Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 7\cdot 37 + 12\cdot 37^{3} + 27\cdot 37^{4} + 15\cdot 37^{5} + 20\cdot 37^{6} + 35\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 30 + \left(14 a + 25\right)\cdot 37 + \left(11 a + 17\right)\cdot 37^{2} + \left(16 a + 1\right)\cdot 37^{3} + \left(3 a + 24\right)\cdot 37^{4} + \left(21 a + 15\right)\cdot 37^{5} + 31 a\cdot 37^{6} + \left(9 a + 21\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 11\cdot 37 + 32\cdot 37^{2} + 35\cdot 37^{3} + 30\cdot 37^{4} + 4\cdot 37^{5} + 21\cdot 37^{6} + 34\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 9 + \left(18 a + 35\right)\cdot 37 + \left(23 a + 4\right)\cdot 37^{2} + \left(22 a + 25\right)\cdot 37^{3} + \left(14 a + 22\right)\cdot 37^{4} + \left(11 a + 10\right)\cdot 37^{5} + \left(9 a + 24\right)\cdot 37^{6} + \left(17 a + 2\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 25 + 22 a\cdot 37 + \left(25 a + 12\right)\cdot 37^{2} + \left(20 a + 18\right)\cdot 37^{3} + \left(33 a + 21\right)\cdot 37^{4} + \left(15 a + 22\right)\cdot 37^{5} + \left(5 a + 31\right)\cdot 37^{6} + \left(27 a + 28\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 25 + \left(18 a + 29\right)\cdot 37 + \left(13 a + 6\right)\cdot 37^{2} + \left(14 a + 18\right)\cdot 37^{3} + \left(22 a + 21\right)\cdot 37^{4} + \left(25 a + 4\right)\cdot 37^{5} + \left(27 a + 13\right)\cdot 37^{6} + \left(19 a + 25\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,5)$ |
| $(1,6)(3,5)$ |
| $(1,5,6,3,4,2)$ |
| $(3,5)(4,6)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $9$ | $2$ | $(1,4)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,6,4)(2,5,3)$ | $-2$ |
| $2$ | $3$ | $(1,6,4)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(2,3,5)$ | $1$ |
| $6$ | $6$ | $(1,5,6,3,4,2)$ | $0$ |
| $6$ | $6$ | $(1,3,6,5,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
