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