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