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:
| $r_{ 1 }$ |
$=$ |
$ a + 5 + \left(11 a + 26\right)\cdot 37 + \left(34 a + 28\right)\cdot 37^{2} + \left(11 a + 8\right)\cdot 37^{3} + \left(14 a + 35\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 23 + \left(a + 31\right)\cdot 37 + \left(4 a + 8\right)\cdot 37^{2} + \left(20 a + 29\right)\cdot 37^{3} + 2 a\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 23 + \left(27 a + 25\right)\cdot 37 + \left(6 a + 21\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(25 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 9 + \left(25 a + 32\right)\cdot 37 + \left(2 a + 6\right)\cdot 37^{2} + \left(25 a + 22\right)\cdot 37^{3} + \left(22 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 9 + \left(35 a + 22\right)\cdot 37 + \left(32 a + 23\right)\cdot 37^{2} + \left(16 a + 31\right)\cdot 37^{3} + \left(34 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 5 + \left(9 a + 10\right)\cdot 37 + \left(30 a + 21\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(11 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,6,5,4,3,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
