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