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:
| $r_{ 1 }$ |
$=$ |
$ 31 + 24\cdot 37 + 4\cdot 37^{2} + 15\cdot 37^{3} + 26\cdot 37^{4} + 6\cdot 37^{5} + 9\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 14 + \left(32 a + 24\right)\cdot 37 + \left(26 a + 9\right)\cdot 37^{2} + \left(14 a + 4\right)\cdot 37^{3} + \left(16 a + 18\right)\cdot 37^{4} + \left(21 a + 20\right)\cdot 37^{5} + \left(a + 29\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 22 + \left(26 a + 17\right)\cdot 37 + \left(5 a + 2\right)\cdot 37^{2} + \left(23 a + 31\right)\cdot 37^{3} + \left(18 a + 12\right)\cdot 37^{4} + \left(24 a + 24\right)\cdot 37^{5} + \left(30 a + 26\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 8 + \left(4 a + 26\right)\cdot 37 + \left(10 a + 10\right)\cdot 37^{2} + \left(22 a + 36\right)\cdot 37^{3} + \left(20 a + 31\right)\cdot 37^{4} + \left(15 a + 15\right)\cdot 37^{5} + \left(35 a + 14\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 23 + \left(10 a + 22\right)\cdot 37 + \left(31 a + 35\right)\cdot 37^{2} + \left(13 a + 6\right)\cdot 37^{3} + \left(18 a + 27\right)\cdot 37^{4} + \left(12 a + 29\right)\cdot 37^{5} + \left(6 a + 13\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 32\cdot 37 + 10\cdot 37^{2} + 17\cdot 37^{3} + 31\cdot 37^{4} + 13\cdot 37^{5} + 17\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,6,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(2,4)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,2,6,4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
