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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 2 + \left(30 a + 8\right)\cdot 37 + 19 a\cdot 37^{2} + \left(35 a + 6\right)\cdot 37^{3} + \left(9 a + 36\right)\cdot 37^{4} + \left(11 a + 34\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 34\cdot 37 + 34\cdot 37^{2} + 8\cdot 37^{3} + 24\cdot 37^{4} + 7\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 35 + \left(10 a + 25\right)\cdot 37 + \left(16 a + 35\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(9 a + 22\right)\cdot 37^{4} + \left(22 a + 25\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 10 + \left(17 a + 16\right)\cdot 37 + \left(3 a + 25\right)\cdot 37^{2} + \left(30 a + 25\right)\cdot 37^{3} + \left(18 a + 33\right)\cdot 37^{4} + \left(21 a + 27\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 16\cdot 37 + 37^{2} + 9\cdot 37^{3} + 13\cdot 37^{4} + 3\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 8 + \left(19 a + 31\right)\cdot 37 + \left(33 a + 21\right)\cdot 37^{2} + \left(6 a + 31\right)\cdot 37^{3} + \left(18 a + 4\right)\cdot 37^{4} + \left(15 a + 21\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a + 21 + \left(26 a + 15\right)\cdot 37 + \left(20 a + 16\right)\cdot 37^{2} + \left(24 a + 4\right)\cdot 37^{3} + \left(27 a + 10\right)\cdot 37^{4} + \left(14 a + 31\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a + 8 + \left(6 a + 36\right)\cdot 37 + \left(17 a + 11\right)\cdot 37^{2} + \left(a + 17\right)\cdot 37^{3} + \left(27 a + 3\right)\cdot 37^{4} + \left(25 a + 33\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(4,8)(6,7)$ |
| $(2,4,7)(5,6,8)$ |
| $(1,4,3,6)(2,7,5,8)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
| $(1,5,3,2)(4,7,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)(7,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(2,5)(4,8)(6,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,7,6)(3,8,4)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,4,3,6)(2,7,5,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5,8,3,2,7)(4,6)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,8,5,4,3,7,2,6)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,5,6,3,8,2,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
