Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{3} + 6 x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a^{2} + 12 a + 1 + \left(24 a^{2} + 8 a + 25\right)\cdot 37 + \left(5 a^{2} + 19 a + 4\right)\cdot 37^{2} + \left(31 a^{2} + 33 a + 1\right)\cdot 37^{3} + \left(25 a^{2} + 11 a + 2\right)\cdot 37^{4} + \left(22 a^{2} + 19 a + 28\right)\cdot 37^{5} + \left(16 a^{2} + 8 a + 34\right)\cdot 37^{6} + \left(35 a^{2} + 5 a + 25\right)\cdot 37^{7} + \left(21 a^{2} + 29 a + 18\right)\cdot 37^{8} + \left(7 a^{2} + 30 a + 6\right)\cdot 37^{9} + \left(17 a^{2} + 24 a + 9\right)\cdot 37^{10} + \left(17 a^{2} + 19 a + 10\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a^{2} + 13 a + 16 + \left(19 a^{2} + 10 a + 2\right)\cdot 37 + \left(12 a^{2} + 32\right)\cdot 37^{2} + \left(28 a^{2} + 4 a + 26\right)\cdot 37^{3} + \left(2 a^{2} + 2 a + 20\right)\cdot 37^{4} + \left(30 a^{2} + 9 a + 20\right)\cdot 37^{5} + \left(24 a + 8\right)\cdot 37^{6} + \left(18 a^{2} + 3 a + 30\right)\cdot 37^{7} + \left(11 a^{2} + 33 a + 13\right)\cdot 37^{8} + \left(12 a^{2} + 18 a + 25\right)\cdot 37^{9} + \left(10 a^{2} + 21 a + 18\right)\cdot 37^{10} + \left(27 a^{2} + 6 a + 12\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a^{2} + 12 a + 33 + \left(30 a^{2} + 18 a + 8\right)\cdot 37 + \left(18 a^{2} + 17 a + 20\right)\cdot 37^{2} + \left(14 a^{2} + 36 a + 8\right)\cdot 37^{3} + \left(8 a^{2} + 22 a + 6\right)\cdot 37^{4} + \left(21 a^{2} + 8 a + 22\right)\cdot 37^{5} + \left(19 a^{2} + 4 a + 9\right)\cdot 37^{6} + \left(20 a^{2} + 28 a + 3\right)\cdot 37^{7} + \left(3 a^{2} + 11 a + 19\right)\cdot 37^{8} + \left(17 a^{2} + 24 a + 7\right)\cdot 37^{9} + \left(9 a^{2} + 27 a + 15\right)\cdot 37^{10} + \left(29 a^{2} + 10 a + 20\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 24\cdot 37 + 6\cdot 37^{2} + 31\cdot 37^{3} + 4\cdot 37^{4} + 24\cdot 37^{6} + 3\cdot 37^{7} + 20\cdot 37^{8} + 23\cdot 37^{9} + 14\cdot 37^{10} + 34\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a^{2} + 35 a + 8 + \left(18 a^{2} + 27 a + 15\right)\cdot 37 + \left(3 a^{2} + 26 a + 2\right)\cdot 37^{2} + \left(15 a^{2} + 18 a + 27\right)\cdot 37^{3} + \left(31 a^{2} + 19 a + 26\right)\cdot 37^{4} + \left(36 a^{2} + 36 a + 33\right)\cdot 37^{5} + \left(24 a^{2} + 15 a + 15\right)\cdot 37^{6} + \left(9 a^{2} + a + 29\right)\cdot 37^{7} + \left(26 a^{2} + 20 a + 17\right)\cdot 37^{8} + \left(19 a^{2} + 34 a + 25\right)\cdot 37^{9} + \left(20 a^{2} + 3 a + 20\right)\cdot 37^{10} + \left(36 a^{2} + 13 a + 20\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a^{2} + 10 a + 10 + \left(36 a^{2} + 13 a + 11\right)\cdot 37 + \left(7 a^{2} + 25 a + 20\right)\cdot 37^{2} + \left(23 a^{2} + 33 a + 22\right)\cdot 37^{3} + \left(19 a^{2} + 22 a + 16\right)\cdot 37^{4} + \left(17 a^{2} + 32 a + 30\right)\cdot 37^{5} + \left(19 a^{2} + 2 a + 30\right)\cdot 37^{6} + \left(26 a^{2} + 36 a + 22\right)\cdot 37^{7} + \left(18 a^{2} + 36 a + 24\right)\cdot 37^{8} + \left(35 a^{2} + 36 a + 14\right)\cdot 37^{9} + \left(11 a^{2} + 25 a + 23\right)\cdot 37^{10} + \left(12 a^{2} + 6 a + 34\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 + 8\cdot 37 + 19\cdot 37^{2} + 22\cdot 37^{3} + 26\cdot 37^{4} + 8\cdot 37^{5} + 17\cdot 37^{6} + 36\cdot 37^{7} + 19\cdot 37^{8} + 32\cdot 37^{9} + 5\cdot 37^{10} + 4\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 + 34\cdot 37 + 25\cdot 37^{2} + 9\cdot 37^{3} + 14\cdot 37^{4} + 2\cdot 37^{5} + 10\cdot 37^{6} + 2\cdot 37^{7} + 22\cdot 37^{8} + 27\cdot 37^{9} + 9\cdot 37^{10} + 36\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 13 a^{2} + 29 a + 13 + \left(19 a^{2} + 32 a + 18\right)\cdot 37 + \left(25 a^{2} + 21 a + 16\right)\cdot 37^{2} + \left(35 a^{2} + 21 a + 35\right)\cdot 37^{3} + \left(22 a^{2} + 31 a + 29\right)\cdot 37^{4} + \left(19 a^{2} + 4 a + 1\right)\cdot 37^{5} + \left(29 a^{2} + 18 a + 34\right)\cdot 37^{6} + \left(36 a + 30\right)\cdot 37^{7} + \left(29 a^{2} + 16 a + 28\right)\cdot 37^{8} + \left(18 a^{2} + 2 a + 21\right)\cdot 37^{9} + \left(4 a^{2} + 7 a + 30\right)\cdot 37^{10} + \left(25 a^{2} + 17 a + 11\right)\cdot 37^{11} +O\left(37^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,7,8)(5,9,6)$ |
| $(4,9)(5,7)(6,8)$ |
| $(1,3,2)(4,7,8)(5,6,9)$ |
| $(1,7,6)(2,4,5)(3,8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,9)(2,6)(3,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,2)(4,7,8)(5,6,9)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,3)(4,8,7)(5,9,6)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $6$ |
$3$ |
$(1,7,6)(2,4,5)(3,8,9)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,8,6)(2,7,5)(3,4,9)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,4,6)(2,8,5)(3,7,9)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(4,7,8)(5,9,6)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,5,2,9,3,6)(4,7,8)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,6,3,9,2,5)(4,8,7)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
