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 + 32 + \left(31 a + 36\right)\cdot 37 + \left(14 a + 31\right)\cdot 37^{2} + \left(3 a + 7\right)\cdot 37^{3} + \left(32 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 36\cdot 37 + 30\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 33 + \left(31 a + 28\right)\cdot 37 + \left(12 a + 35\right)\cdot 37^{2} + \left(13 a + 12\right)\cdot 37^{3} + \left(31 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 36 + \left(5 a + 11\right)\cdot 37 + \left(22 a + 23\right)\cdot 37^{2} + \left(33 a + 6\right)\cdot 37^{3} + \left(4 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 32 + \left(5 a + 33\right)\cdot 37 + \left(24 a + 18\right)\cdot 37^{2} + \left(23 a + 16\right)\cdot 37^{3} + \left(5 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,5)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,3,5,4,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
