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