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