Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 25 + \left(29 a + 30\right)\cdot 37 + \left(20 a + 30\right)\cdot 37^{2} + \left(31 a + 20\right)\cdot 37^{3} + 24\cdot 37^{4} + \left(31 a + 2\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 34 + \left(7 a + 8\right)\cdot 37 + \left(16 a + 10\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(36 a + 33\right)\cdot 37^{4} + \left(5 a + 14\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 34\cdot 37 + 32\cdot 37^{2} + 16\cdot 37^{4} + 19\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 35 + \left(27 a + 36\right)\cdot 37 + \left(32 a + 8\right)\cdot 37^{2} + \left(31 a + 4\right)\cdot 37^{3} + \left(33 a + 32\right)\cdot 37^{4} + \left(13 a + 4\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 18 + \left(9 a + 29\right)\cdot 37 + \left(4 a + 1\right)\cdot 37^{2} + \left(5 a + 25\right)\cdot 37^{3} + \left(3 a + 24\right)\cdot 37^{4} + \left(23 a + 26\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 7\cdot 37 + 26\cdot 37^{2} + 7\cdot 37^{3} + 17\cdot 37^{4} + 5\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)(2,5)(3,6)$ |
| $(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,4,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
