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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 29\cdot 37 + 15\cdot 37^{2} + 8\cdot 37^{3} + 33\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 36\cdot 37 + 27\cdot 37^{2} + 28\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + \left(20 a + 30\right)\cdot 37 + \left(14 a + 28\right)\cdot 37^{2} + \left(8 a + 24\right)\cdot 37^{3} + \left(4 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 31 + \left(16 a + 20\right)\cdot 37 + \left(22 a + 29\right)\cdot 37^{2} + \left(28 a + 6\right)\cdot 37^{3} + \left(32 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 27\cdot 37 + 37^{2} + 16\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 3\cdot 37 + 7\cdot 37^{2} + 26\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
