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:
| $r_{ 1 }$ |
$=$ |
$ 32 + 15\cdot 37 + 31\cdot 37^{2} + 36\cdot 37^{3} + 26\cdot 37^{4} + 12\cdot 37^{5} + 26\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 22 + 37 + \left(8 a + 26\right)\cdot 37^{2} + \left(28 a + 2\right)\cdot 37^{3} + \left(30 a + 34\right)\cdot 37^{4} + \left(25 a + 18\right)\cdot 37^{5} + \left(4 a + 5\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 31 + 27\cdot 37 + \left(8 a + 16\right)\cdot 37^{2} + \left(28 a + 3\right)\cdot 37^{3} + \left(30 a + 19\right)\cdot 37^{4} + \left(25 a + 19\right)\cdot 37^{5} + \left(4 a + 1\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 + 21\cdot 37 + 5\cdot 37^{2} + 10\cdot 37^{4} + 24\cdot 37^{5} + 10\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 15 + \left(36 a + 35\right)\cdot 37 + \left(28 a + 10\right)\cdot 37^{2} + \left(8 a + 34\right)\cdot 37^{3} + \left(6 a + 2\right)\cdot 37^{4} + \left(11 a + 18\right)\cdot 37^{5} + \left(32 a + 31\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 6 + \left(36 a + 9\right)\cdot 37 + \left(28 a + 20\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(6 a + 17\right)\cdot 37^{4} + \left(11 a + 17\right)\cdot 37^{5} + \left(32 a + 35\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,3)(2,5)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
