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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 18 + \left(33 a + 24\right)\cdot 37 + \left(9 a + 14\right)\cdot 37^{2} + \left(32 a + 11\right)\cdot 37^{3} + \left(23 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 20 + \left(36 a + 16\right)\cdot 37 + \left(35 a + 23\right)\cdot 37^{2} + \left(33 a + 33\right)\cdot 37^{3} + \left(5 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 8 + 19\cdot 37 + \left(a + 19\right)\cdot 37^{2} + \left(3 a + 22\right)\cdot 37^{3} + \left(31 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a + 33 + \left(33 a + 20\right)\cdot 37 + \left(8 a + 33\right)\cdot 37^{2} + \left(29 a + 31\right)\cdot 37^{3} + \left(29 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 36 + \left(3 a + 32\right)\cdot 37 + \left(28 a + 35\right)\cdot 37^{2} + \left(7 a + 28\right)\cdot 37^{3} + \left(7 a + 3\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 33 + \left(3 a + 33\right)\cdot 37 + \left(27 a + 20\right)\cdot 37^{2} + \left(4 a + 19\right)\cdot 37^{3} + \left(13 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,5,2)(3,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,6,5,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,6,2,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
