Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 3 + \left(35 a + 26\right)\cdot 37 + \left(24 a + 30\right)\cdot 37^{2} + \left(33 a + 20\right)\cdot 37^{3} + \left(a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 24\cdot 37 + 22\cdot 37^{2} + 33\cdot 37^{3} + 3\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 23 + \left(26 a + 2\right)\cdot 37 + \left(19 a + 6\right)\cdot 37^{2} + \left(28 a + 9\right)\cdot 37^{3} + \left(20 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 22\cdot 37 + 9\cdot 37^{2} + 35\cdot 37^{3} + 6\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 32 + \left(a + 23\right)\cdot 37 + \left(12 a + 20\right)\cdot 37^{2} + \left(3 a + 19\right)\cdot 37^{3} + \left(35 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 4 + \left(10 a + 12\right)\cdot 37 + \left(17 a + 21\right)\cdot 37^{2} + \left(8 a + 29\right)\cdot 37^{3} + \left(16 a + 23\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,2)$ | $0$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(3,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,5,6)$ | $-1$ |
| $12$ | $6$ | $(1,2)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
