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