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