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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 34\cdot 37^{2} + 11\cdot 37^{3} + 15\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 29 + \left(4 a + 30\right)\cdot 37 + \left(6 a + 24\right)\cdot 37^{2} + \left(7 a + 3\right)\cdot 37^{3} + \left(32 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 10 + \left(32 a + 26\right)\cdot 37 + \left(30 a + 7\right)\cdot 37^{2} + \left(29 a + 26\right)\cdot 37^{3} + \left(4 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + \left(9 a + 15\right)\cdot 37 + \left(35 a + 30\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(3 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 19 + \left(27 a + 1\right)\cdot 37 + \left(a + 14\right)\cdot 37^{2} + \left(24 a + 24\right)\cdot 37^{3} + \left(33 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
