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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 36 + \left(16 a + 22\right)\cdot 37 + \left(21 a + 11\right)\cdot 37^{2} + \left(24 a + 32\right)\cdot 37^{3} + \left(26 a + 7\right)\cdot 37^{4} + \left(16 a + 29\right)\cdot 37^{5} + \left(18 a + 12\right)\cdot 37^{6} + \left(6 a + 5\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 5 + \left(21 a + 7\right)\cdot 37 + \left(33 a + 12\right)\cdot 37^{2} + \left(28 a + 25\right)\cdot 37^{3} + \left(29 a + 7\right)\cdot 37^{4} + \left(21 a + 30\right)\cdot 37^{5} + \left(11 a + 22\right)\cdot 37^{6} + \left(22 a + 3\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 13 + \left(20 a + 30\right)\cdot 37 + \left(15 a + 6\right)\cdot 37^{2} + \left(12 a + 35\right)\cdot 37^{3} + \left(10 a + 15\right)\cdot 37^{4} + \left(20 a + 32\right)\cdot 37^{5} + \left(18 a + 32\right)\cdot 37^{6} + \left(30 a + 12\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 20\cdot 37 + 18\cdot 37^{2} + 6\cdot 37^{3} + 13\cdot 37^{4} + 12\cdot 37^{5} + 28\cdot 37^{6} + 18\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 26 + \left(15 a + 24\right)\cdot 37 + \left(3 a + 13\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(7 a + 23\right)\cdot 37^{4} + \left(15 a + 13\right)\cdot 37^{5} + \left(25 a + 10\right)\cdot 37^{6} + \left(14 a + 7\right)\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 5\cdot 37 + 11\cdot 37^{2} + 15\cdot 37^{3} + 5\cdot 37^{4} + 30\cdot 37^{5} + 3\cdot 37^{6} + 26\cdot 37^{7} +O\left(37^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)(2,5)$ |
| $(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
