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