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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 35 + \left(22 a + 13\right)\cdot 37 + \left(21 a + 1\right)\cdot 37^{2} + \left(32 a + 13\right)\cdot 37^{3} + \left(31 a + 10\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 17 + \left(14 a + 15\right)\cdot 37 + \left(15 a + 28\right)\cdot 37^{2} + \left(4 a + 10\right)\cdot 37^{3} + \left(5 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 33 + 7\cdot 37 + \left(2 a + 25\right)\cdot 37^{2} + \left(17 a + 34\right)\cdot 37^{3} + \left(26 a + 4\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 24 + \left(21 a + 13\right)\cdot 37 + \left(19 a + 20\right)\cdot 37^{2} + \left(15 a + 28\right)\cdot 37^{3} + 5 a\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 31 + \left(36 a + 28\right)\cdot 37 + \left(34 a + 32\right)\cdot 37^{2} + \left(19 a + 26\right)\cdot 37^{3} + \left(10 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 8 + \left(15 a + 31\right)\cdot 37 + \left(17 a + 2\right)\cdot 37^{2} + \left(21 a + 34\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(1,2,6,4,5,3)$ |
| $(1,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,6,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
| $3$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
