Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 11\cdot 37 + 35\cdot 37^{2} + 4\cdot 37^{3} + 32\cdot 37^{4} + 9\cdot 37^{5} + 20\cdot 37^{6} + 6\cdot 37^{7} + 32\cdot 37^{8} + 24\cdot 37^{9} + 5\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 22 + \left(18 a + 28\right)\cdot 37 + \left(23 a + 6\right)\cdot 37^{2} + \left(3 a + 16\right)\cdot 37^{3} + \left(8 a + 7\right)\cdot 37^{4} + 27\cdot 37^{5} + \left(2 a + 4\right)\cdot 37^{6} + \left(23 a + 30\right)\cdot 37^{7} + \left(3 a + 17\right)\cdot 37^{8} + \left(24 a + 18\right)\cdot 37^{9} + \left(22 a + 16\right)\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 35 + \left(18 a + 35\right)\cdot 37 + \left(13 a + 7\right)\cdot 37^{2} + \left(33 a + 7\right)\cdot 37^{3} + \left(28 a + 36\right)\cdot 37^{4} + \left(36 a + 19\right)\cdot 37^{5} + \left(34 a + 12\right)\cdot 37^{6} + \left(13 a + 9\right)\cdot 37^{7} + \left(33 a + 9\right)\cdot 37^{8} + 12 a\cdot 37^{9} + \left(14 a + 9\right)\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 26\cdot 37 + 18\cdot 37^{2} + 24\cdot 37^{3} + 2\cdot 37^{4} + 29\cdot 37^{5} + 19\cdot 37^{6} + 33\cdot 37^{7} + 22\cdot 37^{8} + 12\cdot 37^{9} + 21\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 4 + \left(2 a + 10\right)\cdot 37 + \left(26 a + 7\right)\cdot 37^{2} + \left(7 a + 8\right)\cdot 37^{3} + \left(4 a + 30\right)\cdot 37^{4} + \left(12 a + 8\right)\cdot 37^{5} + \left(36 a + 34\right)\cdot 37^{6} + \left(6 a + 19\right)\cdot 37^{7} + \left(15 a + 24\right)\cdot 37^{8} + \left(7 a + 1\right)\cdot 37^{9} + \left(35 a + 18\right)\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 22 + \left(34 a + 34\right)\cdot 37 + \left(10 a + 34\right)\cdot 37^{2} + \left(29 a + 12\right)\cdot 37^{3} + \left(32 a + 2\right)\cdot 37^{4} + \left(24 a + 16\right)\cdot 37^{5} + 19\cdot 37^{6} + \left(30 a + 11\right)\cdot 37^{7} + \left(21 a + 4\right)\cdot 37^{8} + \left(29 a + 16\right)\cdot 37^{9} + \left(a + 3\right)\cdot 37^{10} +O\left(37^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,5,2)(3,4,6)$ |
| $(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(2,3)(5,6)$ |
$-1$ |
| $3$ |
$2$ |
$(2,3)$ |
$1$ |
| $6$ |
$2$ |
$(2,5)(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,5,2)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(2,6,3,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,4)(2,6,3,5)$ |
$-1$ |
| $8$ |
$6$ |
$(1,2,6,4,3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
