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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + \left(19 a + 6\right)\cdot 37 + \left(14 a + 34\right)\cdot 37^{2} + \left(2 a + 12\right)\cdot 37^{3} + \left(12 a + 36\right)\cdot 37^{4} + \left(5 a + 4\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 15 + \left(17 a + 33\right)\cdot 37 + \left(22 a + 35\right)\cdot 37^{2} + \left(34 a + 7\right)\cdot 37^{3} + \left(24 a + 8\right)\cdot 37^{4} + \left(31 a + 14\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 4 + \left(4 a + 30\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(28 a + 27\right)\cdot 37^{3} + \left(19 a + 33\right)\cdot 37^{4} + \left(8 a + 2\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 8 + 22 a\cdot 37 + \left(23 a + 21\right)\cdot 37^{2} + \left(20 a + 24\right)\cdot 37^{3} + \left(30 a + 28\right)\cdot 37^{4} + \left(34 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 37^{2} + 23\cdot 37^{3} + 19\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 11 + \left(32 a + 36\right)\cdot 37 + \left(35 a + 7\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(17 a + 10\right)\cdot 37^{4} + \left(28 a + 17\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 11 + 31\cdot 37 + 20\cdot 37^{2} + 15\cdot 37^{3} + 28\cdot 37^{4} + 4\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 32 a + 28 + \left(14 a + 9\right)\cdot 37 + \left(13 a + 19\right)\cdot 37^{2} + \left(16 a + 9\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} + \left(2 a + 6\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,2,8)(3,5,4,7)$ |
| $(1,6,5)(2,8,7)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,7,2,5)(3,6,4,8)$ |
| $(1,2)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,2)(5,8)(6,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,8,3)(2,6,4)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,2,5)(3,6,4,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,4,8,2,3,6)(5,7)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,5,3,6,2,7,4,8)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,3,8,2,5,4,6)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
