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