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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 21 + \left(11 a + 25\right)\cdot 37 + \left(26 a + 31\right)\cdot 37^{2} + \left(3 a + 18\right)\cdot 37^{3} + \left(31 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 31 + \left(25 a + 32\right)\cdot 37 + \left(10 a + 3\right)\cdot 37^{2} + \left(33 a + 12\right)\cdot 37^{3} + \left(5 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 2 + \left(11 a + 21\right)\cdot 37 + \left(26 a + 28\right)\cdot 37^{2} + \left(3 a + 11\right)\cdot 37^{3} + \left(31 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 27 + \left(25 a + 14\right)\cdot 37 + \left(10 a + 14\right)\cdot 37^{2} + \left(33 a + 7\right)\cdot 37^{3} + \left(5 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 25 + \left(11 a + 6\right)\cdot 37 + \left(26 a + 21\right)\cdot 37^{2} + \left(3 a + 23\right)\cdot 37^{3} + \left(31 a + 28\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 8 + \left(25 a + 10\right)\cdot 37 + \left(10 a + 11\right)\cdot 37^{2} + 33 a\cdot 37^{3} + 5 a\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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
