Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 35\cdot 37 + 23\cdot 37^{2} + 2\cdot 37^{3} + 14\cdot 37^{4} + 25\cdot 37^{5} + 7\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 20 + \left(10 a + 11\right)\cdot 37 + \left(14 a + 12\right)\cdot 37^{2} + \left(13 a + 1\right)\cdot 37^{3} + \left(9 a + 29\right)\cdot 37^{4} + \left(6 a + 21\right)\cdot 37^{5} + \left(22 a + 7\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 7 + \left(35 a + 31\right)\cdot 37 + \left(11 a + 25\right)\cdot 37^{2} + \left(8 a + 16\right)\cdot 37^{3} + \left(33 a + 17\right)\cdot 37^{4} + \left(2 a + 14\right)\cdot 37^{5} + \left(23 a + 26\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 30 + \left(26 a + 32\right)\cdot 37 + \left(22 a + 21\right)\cdot 37^{2} + \left(23 a + 3\right)\cdot 37^{3} + \left(27 a + 16\right)\cdot 37^{4} + 30 a\cdot 37^{5} + \left(14 a + 16\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 27 + \left(a + 18\right)\cdot 37 + \left(25 a + 1\right)\cdot 37^{2} + \left(28 a + 1\right)\cdot 37^{3} + \left(3 a + 31\right)\cdot 37^{4} + \left(34 a + 29\right)\cdot 37^{5} + \left(13 a + 4\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 18\cdot 37 + 25\cdot 37^{2} + 11\cdot 37^{3} + 3\cdot 37^{4} + 19\cdot 37^{5} + 11\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,5,6)$ |
| $(2,3)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,4)(3,5)$ | $1$ |
| $8$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,6,2)$ | $-1$ |
| $6$ | $4$ | $(1,5,6,4)(2,3)$ | $1$ |
| $8$ | $6$ | $(1,3,5,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
