Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 30 + \left(33 a + 10\right)\cdot 37 + \left(10 a + 25\right)\cdot 37^{2} + \left(30 a + 24\right)\cdot 37^{3} + \left(8 a + 16\right)\cdot 37^{4} + \left(26 a + 13\right)\cdot 37^{5} + \left(17 a + 34\right)\cdot 37^{6} + \left(34 a + 1\right)\cdot 37^{7} + \left(36 a + 22\right)\cdot 37^{8} + \left(11 a + 32\right)\cdot 37^{9} + \left(8 a + 4\right)\cdot 37^{10} + \left(7 a + 32\right)\cdot 37^{11} + \left(7 a + 11\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 10 + \left(18 a + 2\right)\cdot 37 + \left(22 a + 17\right)\cdot 37^{2} + \left(31 a + 11\right)\cdot 37^{3} + \left(20 a + 24\right)\cdot 37^{4} + \left(22 a + 22\right)\cdot 37^{5} + \left(34 a + 28\right)\cdot 37^{6} + \left(4 a + 18\right)\cdot 37^{7} + 30\cdot 37^{8} + \left(8 a + 7\right)\cdot 37^{9} + \left(3 a + 27\right)\cdot 37^{10} + \left(5 a + 33\right)\cdot 37^{11} + \left(28 a + 14\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 18\cdot 37 + 23\cdot 37^{2} + 7\cdot 37^{4} + 25\cdot 37^{5} + 7\cdot 37^{6} + 2\cdot 37^{7} + 3\cdot 37^{8} + 33\cdot 37^{9} + 8\cdot 37^{10} + 27\cdot 37^{11} + 22\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 32\cdot 37^{2} + 8\cdot 37^{3} + 2\cdot 37^{4} + 33\cdot 37^{5} + 37^{6} + 36\cdot 37^{7} + 4\cdot 37^{8} + 28\cdot 37^{9} + 11\cdot 37^{10} + 25\cdot 37^{11} + 16\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 12 + \left(18 a + 20\right)\cdot 37 + \left(14 a + 14\right)\cdot 37^{2} + \left(5 a + 4\right)\cdot 37^{3} + \left(16 a + 2\right)\cdot 37^{4} + \left(14 a + 18\right)\cdot 37^{5} + \left(2 a + 33\right)\cdot 37^{6} + \left(32 a + 3\right)\cdot 37^{7} + \left(36 a + 26\right)\cdot 37^{8} + \left(28 a + 2\right)\cdot 37^{9} + \left(33 a + 32\right)\cdot 37^{10} + \left(31 a + 13\right)\cdot 37^{11} + \left(8 a + 11\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 4 + \left(3 a + 21\right)\cdot 37 + \left(26 a + 35\right)\cdot 37^{2} + \left(6 a + 23\right)\cdot 37^{3} + \left(28 a + 21\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + \left(19 a + 4\right)\cdot 37^{6} + \left(2 a + 11\right)\cdot 37^{7} + 24\cdot 37^{8} + \left(25 a + 6\right)\cdot 37^{9} + \left(28 a + 26\right)\cdot 37^{10} + \left(29 a + 15\right)\cdot 37^{11} + \left(29 a + 33\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(1,5,3)(2,6,4)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(5,6)$ |
$1$ |
| $3$ |
$2$ |
$(3,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,3)(2,4)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$0$ |
| $6$ |
$4$ |
$(3,5,4,6)$ |
$1$ |
| $6$ |
$4$ |
$(1,2)(3,5,4,6)$ |
$-1$ |
| $8$ |
$6$ |
$(1,5,4,2,6,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
