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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 22\cdot 37 + 37^{2} + 15\cdot 37^{3} + 14\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 13\cdot 37 + 4\cdot 37^{2} + 9\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 32 + \left(9 a + 1\right)\cdot 37 + \left(12 a + 28\right)\cdot 37^{2} + \left(34 a + 30\right)\cdot 37^{3} + \left(33 a + 1\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 13 + \left(27 a + 17\right)\cdot 37 + \left(24 a + 30\right)\cdot 37^{2} + \left(2 a + 7\right)\cdot 37^{3} + \left(3 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 34 + \left(29 a + 34\right)\cdot 37 + \left(20 a + 14\right)\cdot 37^{2} + \left(30 a + 10\right)\cdot 37^{3} + \left(31 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 7 + \left(7 a + 21\right)\cdot 37 + \left(16 a + 31\right)\cdot 37^{2} + 6 a\cdot 37^{3} + \left(5 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,6,2)(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $20$ | $3$ | $(1,2,6)(3,5,4)$ | $0$ |
| $12$ | $5$ | $(1,3,4,2,6)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,4,6,3,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
