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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 20\cdot 37 + 10\cdot 37^{2} + 21\cdot 37^{3} + 4\cdot 37^{4} + 28\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 24 + \left(28 a + 23\right)\cdot 37 + \left(8 a + 6\right)\cdot 37^{2} + \left(30 a + 15\right)\cdot 37^{3} + \left(6 a + 19\right)\cdot 37^{4} + \left(3 a + 14\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 4 + \left(8 a + 33\right)\cdot 37 + \left(28 a + 12\right)\cdot 37^{2} + \left(6 a + 16\right)\cdot 37^{3} + \left(30 a + 16\right)\cdot 37^{4} + \left(33 a + 20\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 36 + \left(32 a + 26\right)\cdot 37 + \left(24 a + 25\right)\cdot 37^{2} + \left(22 a + 8\right)\cdot 37^{3} + \left(5 a + 10\right)\cdot 37^{4} + \left(21 a + 15\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 21\cdot 37 + 36\cdot 37^{2} + 11\cdot 37^{3} + 13\cdot 37^{4} + 12\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 21 + \left(4 a + 22\right)\cdot 37 + \left(12 a + 18\right)\cdot 37^{2} + 14 a\cdot 37^{3} + \left(31 a + 10\right)\cdot 37^{4} + \left(15 a + 20\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(4,5,6)$ |
| $(2,3)(5,6)$ |
| $(1,6,2,5,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$0$ |
| $9$ |
$2$ |
$(1,3)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$-2$ |
| $2$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $4$ |
$3$ |
$(4,5,6)$ |
$1$ |
| $6$ |
$6$ |
$(1,6,2,5,3,4)$ |
$0$ |
| $6$ |
$6$ |
$(1,5,2,6,3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
