The Arithmetic of Quaternion Algebra

5.3. EXAMPLES 117
The class number of ideals for the relation J = aIb, I, J are the ideals of the
order of level N, a, b ∈ H × is given by the formula
h + �
−r
= 2
m|DN
tr(m) 2 .
these tables were computed by Henri Cohen of The Center of Computation at
Bordeaux.
Here are tables occupying two pages!!! See this original book, pp.153,154.
With the help of these tables, we can prove that there are 10 Eichler orders
of the level N without square factors , of a quaternion field totally defined
over Q, and of the reduced discriminant D, and of the class number 1 ( up to
isomorphisms). We obtain them with:
D N
2 1,3,5,11
3 1,2
5 1,2
7 1
13 1
The explicit computation for the quaternion algebra totally defined over a
real quadratic field Q( √ m) Allows to prove that the Eichler orders of level N
without square factor of the quaternion algebras , which are totally defined over
Q( √ m) of the reduced discriminant D, have the class number equal to h + m, and
have the class number in the restrict sense of Q( √ m), are obtained with the
following data:
m D N
2 1,p2p3, p2p5, p2p (i)
7
1 p2, p (i)
7
3 p2p3, p2p5, p2p (i) (i)
13 , p3p 13
1
1
, p(i)
23
1 p2, p3, p (i)
11
5 1,p2p5, p2p (i)
11
1 p2, p3, p5, p (i)
11
6 p2p3, p3p (i)
5
13 1, p2p (i)
3
1
1
1
1 p (i)
5
15 p2p3 1
17 1 1, p (i)
2
21 1, p2p3 1
1 p (i)
5
33 p (i)
2 p3 1
, p(i)
19
, p(i) 29 , p(i) 59
There are 54 couples (D, N). The ideals p (i)
a , i = 1, 2 represent the prime ideals