Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 17 + \left(7 a + 27\right)\cdot 37 + \left(3 a + 15\right)\cdot 37^{2} + \left(28 a + 19\right)\cdot 37^{3} + 12\cdot 37^{4} + \left(17 a + 3\right)\cdot 37^{5} + \left(6 a + 14\right)\cdot 37^{6} + \left(33 a + 29\right)\cdot 37^{7} + \left(23 a + 5\right)\cdot 37^{8} + \left(36 a + 31\right)\cdot 37^{9} + \left(2 a + 30\right)\cdot 37^{10} + \left(26 a + 4\right)\cdot 37^{11} + \left(20 a + 27\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 30 + \left(18 a + 10\right)\cdot 37 + \left(7 a + 31\right)\cdot 37^{2} + 21\cdot 37^{3} + \left(28 a + 36\right)\cdot 37^{4} + \left(11 a + 8\right)\cdot 37^{5} + \left(16 a + 10\right)\cdot 37^{6} + \left(30 a + 21\right)\cdot 37^{7} + \left(12 a + 26\right)\cdot 37^{8} + \left(2 a + 1\right)\cdot 37^{9} + \left(4 a + 30\right)\cdot 37^{10} + \left(36 a + 3\right)\cdot 37^{11} + \left(33 a + 24\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 22\cdot 37 + 27\cdot 37^{2} + 13\cdot 37^{3} + 33\cdot 37^{4} + 27\cdot 37^{5} + 23\cdot 37^{6} + 17\cdot 37^{8} + 13\cdot 37^{9} + 37^{10} + 31\cdot 37^{11} + 35\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 20 + \left(29 a + 9\right)\cdot 37 + \left(33 a + 21\right)\cdot 37^{2} + \left(8 a + 17\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} + \left(19 a + 33\right)\cdot 37^{5} + \left(30 a + 22\right)\cdot 37^{6} + \left(3 a + 7\right)\cdot 37^{7} + \left(13 a + 31\right)\cdot 37^{8} + 5\cdot 37^{9} + \left(34 a + 6\right)\cdot 37^{10} + \left(10 a + 32\right)\cdot 37^{11} + \left(16 a + 9\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 7 + \left(18 a + 26\right)\cdot 37 + \left(29 a + 5\right)\cdot 37^{2} + \left(36 a + 15\right)\cdot 37^{3} + 8 a\cdot 37^{4} + \left(25 a + 28\right)\cdot 37^{5} + \left(20 a + 26\right)\cdot 37^{6} + \left(6 a + 15\right)\cdot 37^{7} + \left(24 a + 10\right)\cdot 37^{8} + \left(34 a + 35\right)\cdot 37^{9} + \left(32 a + 6\right)\cdot 37^{10} + 33\cdot 37^{11} + \left(3 a + 12\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 14\cdot 37 + 9\cdot 37^{2} + 23\cdot 37^{3} + 3\cdot 37^{4} + 9\cdot 37^{5} + 13\cdot 37^{6} + 36\cdot 37^{7} + 19\cdot 37^{8} + 23\cdot 37^{9} + 35\cdot 37^{10} + 5\cdot 37^{11} + 37^{12} +O\left(37^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
