Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 36 + 4 a\cdot 37 + \left(9 a + 9\right)\cdot 37^{2} + \left(14 a + 25\right)\cdot 37^{3} + \left(26 a + 32\right)\cdot 37^{4} + \left(28 a + 11\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 24 + \left(15 a + 20\right)\cdot 37 + \left(28 a + 5\right)\cdot 37^{2} + \left(7 a + 32\right)\cdot 37^{3} + \left(29 a + 4\right)\cdot 37^{4} + \left(26 a + 11\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 1 + \left(26 a + 27\right)\cdot 37 + \left(17 a + 27\right)\cdot 37^{2} + \left(6 a + 5\right)\cdot 37^{3} + \left(34 a + 19\right)\cdot 37^{4} + \left(a + 36\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 12 + \left(32 a + 26\right)\cdot 37 + \left(27 a + 3\right)\cdot 37^{2} + \left(22 a + 36\right)\cdot 37^{3} + \left(10 a + 12\right)\cdot 37^{4} + \left(8 a + 26\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 25 + \left(21 a + 18\right)\cdot 37 + \left(8 a + 29\right)\cdot 37^{2} + \left(29 a + 34\right)\cdot 37^{3} + \left(7 a + 2\right)\cdot 37^{4} + \left(10 a + 15\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 13 + \left(10 a + 17\right)\cdot 37 + \left(19 a + 35\right)\cdot 37^{2} + \left(30 a + 13\right)\cdot 37^{3} + \left(2 a + 1\right)\cdot 37^{4} + \left(35 a + 10\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,3)$ |
| $(1,3)(2,6)(4,5)$ |
| $(1,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,4,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,3,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,3,6,2,5,4)$ | $0$ |
| $3$ | $6$ | $(1,4,5,2,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
