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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 27 + \left(30 a + 22\right)\cdot 37 + \left(13 a + 10\right)\cdot 37^{2} + \left(12 a + 28\right)\cdot 37^{3} + \left(27 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 16 + \left(6 a + 9\right)\cdot 37 + \left(23 a + 35\right)\cdot 37^{2} + \left(24 a + 26\right)\cdot 37^{3} + \left(9 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 22\cdot 37 + 4\cdot 37^{2} + 19\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 36 + \left(16 a + 17\right)\cdot 37 + \left(23 a + 33\right)\cdot 37^{2} + \left(30 a + 36\right)\cdot 37^{3} + \left(35 a + 34\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 35 + \left(20 a + 36\right)\cdot 37 + \left(13 a + 36\right)\cdot 37^{2} + \left(6 a + 24\right)\cdot 37^{3} + \left(a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 37 + 27\cdot 37^{2} + 11\cdot 37^{3} + 21\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,4,5,3,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,4)(3,5)$ | $-2$ |
| $20$ | $3$ | $(1,5,6)(2,4,3)$ | $0$ |
| $30$ | $4$ | $(1,3,4,5)$ | $0$ |
| $24$ | $5$ | $(2,5,3,6,4)$ | $1$ |
| $20$ | $6$ | $(1,4,5,3,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
