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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 36 + \left(36 a + 22\right)\cdot 37 + 30\cdot 37^{2} + \left(8 a + 22\right)\cdot 37^{3} + \left(36 a + 17\right)\cdot 37^{4} + \left(20 a + 6\right)\cdot 37^{5} + \left(26 a + 32\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 27 + 12\cdot 37 + \left(36 a + 35\right)\cdot 37^{2} + \left(28 a + 16\right)\cdot 37^{3} + 6\cdot 37^{4} + \left(16 a + 17\right)\cdot 37^{5} + \left(10 a + 6\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 32\cdot 37 + 4\cdot 37^{2} + 22\cdot 37^{3} + 2\cdot 37^{4} + 4\cdot 37^{5} + 8\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 3 + \left(31 a + 24\right)\cdot 37 + \left(16 a + 11\right)\cdot 37^{2} + \left(15 a + 20\right)\cdot 37^{3} + \left(28 a + 7\right)\cdot 37^{4} + \left(11 a + 26\right)\cdot 37^{5} + \left(11 a + 13\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 + 12\cdot 37 + 18\cdot 37^{2} + 8\cdot 37^{4} + 12\cdot 37^{5} + 3\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 32 + \left(5 a + 5\right)\cdot 37 + \left(20 a + 10\right)\cdot 37^{2} + \left(21 a + 28\right)\cdot 37^{3} + \left(8 a + 31\right)\cdot 37^{4} + \left(25 a + 7\right)\cdot 37^{5} + \left(25 a + 10\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)(4,6,5)$ |
| $(1,3)(4,5)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,4)(3,5)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,4,3)$ | $-1$ |
| $6$ | $4$ | $(1,5,4,3)(2,6)$ | $1$ |
| $8$ | $6$ | $(1,5,6,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
