Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 12 + 20 a\cdot 37 + \left(8 a + 6\right)\cdot 37^{2} + \left(36 a + 21\right)\cdot 37^{3} + \left(8 a + 29\right)\cdot 37^{4} + \left(17 a + 36\right)\cdot 37^{5} + \left(7 a + 5\right)\cdot 37^{6} + \left(19 a + 34\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 26\cdot 37 + 18\cdot 37^{2} + 5\cdot 37^{3} + 11\cdot 37^{4} + 12\cdot 37^{5} + 2\cdot 37^{6} + 8\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 29 + \left(36 a + 10\right)\cdot 37 + \left(31 a + 26\right)\cdot 37^{2} + \left(8 a + 29\right)\cdot 37^{3} + 6 a\cdot 37^{4} + \left(4 a + 9\right)\cdot 37^{5} + 30 a\cdot 37^{6} + \left(21 a + 20\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 30 + 18\cdot 37 + \left(5 a + 6\right)\cdot 37^{2} + \left(28 a + 33\right)\cdot 37^{3} + \left(30 a + 16\right)\cdot 37^{4} + \left(32 a + 19\right)\cdot 37^{5} + \left(6 a + 5\right)\cdot 37^{6} + \left(15 a + 3\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 6\cdot 37 + 34\cdot 37^{2} + 11\cdot 37^{3} + 23\cdot 37^{4} + 10\cdot 37^{5} + 4\cdot 37^{6} + 16\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 8 + \left(16 a + 11\right)\cdot 37 + \left(28 a + 19\right)\cdot 37^{2} + 9\cdot 37^{3} + \left(28 a + 29\right)\cdot 37^{4} + \left(19 a + 22\right)\cdot 37^{5} + \left(29 a + 18\right)\cdot 37^{6} + \left(17 a + 29\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(3,5)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,3,2)$ | $1$ |
| $6$ | $4$ | $(1,3)(2,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,2,3,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
