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