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