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:
| $r_{ 1 }$ |
$=$ |
$ 15 + 12\cdot 37 + 21\cdot 37^{2} + 4\cdot 37^{3} + 33\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 6 + \left(24 a + 36\right)\cdot 37 + 36\cdot 37^{2} + \left(21 a + 22\right)\cdot 37^{3} + \left(30 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 9 + \left(12 a + 12\right)\cdot 37 + \left(36 a + 15\right)\cdot 37^{2} + \left(15 a + 32\right)\cdot 37^{3} + \left(6 a + 12\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 13 + \left(36 a + 28\right)\cdot 37 + \left(20 a + 20\right)\cdot 37^{2} + 13 a\cdot 37^{3} + \left(17 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 20 + 14\cdot 37 + \left(16 a + 31\right)\cdot 37^{2} + \left(23 a + 33\right)\cdot 37^{3} + \left(19 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 7\cdot 37 + 22\cdot 37^{2} + 16\cdot 37^{3} + 7\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,6,2,3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $15$ | $2$ | $(2,6)(3,5)$ | $0$ |
| $20$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $30$ | $4$ | $(2,3,6,5)$ | $0$ |
| $24$ | $5$ | $(1,3,5,4,2)$ | $-1$ |
| $20$ | $6$ | $(1,6,2,3,5,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
