Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 3 + \left(2 a + 17\right)\cdot 37 + \left(7 a + 27\right)\cdot 37^{2} + \left(14 a + 18\right)\cdot 37^{3} + \left(29 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 16 + \left(34 a + 34\right)\cdot 37 + \left(29 a + 15\right)\cdot 37^{2} + \left(22 a + 31\right)\cdot 37^{3} + \left(7 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 6\cdot 37 + 29\cdot 37^{2} + 14\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 16 + \left(21 a + 13\right)\cdot 37 + \left(6 a + 1\right)\cdot 37^{2} + 7 a\cdot 37^{3} + \left(18 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 7 + \left(15 a + 17\right)\cdot 37 + \left(30 a + 6\right)\cdot 37^{2} + \left(29 a + 22\right)\cdot 37^{3} + \left(18 a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 22\cdot 37 + 30\cdot 37^{2} + 23\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(4,5)$ | $0$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(3,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,4,6,5)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
| $12$ | $6$ | $(1,2,6)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
