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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 33 + \left(22 a + 8\right)\cdot 37 + \left(21 a + 6\right)\cdot 37^{2} + \left(19 a + 25\right)\cdot 37^{3} + \left(35 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 11 + 3 a\cdot 37 + \left(7 a + 17\right)\cdot 37^{2} + \left(30 a + 36\right)\cdot 37^{3} + \left(32 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 28 + \left(33 a + 20\right)\cdot 37 + \left(29 a + 4\right)\cdot 37^{2} + \left(6 a + 2\right)\cdot 37^{3} + \left(4 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 16 + \left(14 a + 18\right)\cdot 37 + \left(15 a + 33\right)\cdot 37^{2} + \left(17 a + 7\right)\cdot 37^{3} + \left(a + 11\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 25\cdot 37 + 12\cdot 37^{2} + 2\cdot 37^{3} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
