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