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