# Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

The coincidence part of this is Jyrki bringing up this idea within a couple of weeks of Neil Sloane asking about programming for indefinite forms....

However, I remain a little uncertain about how this works with indefinite forms...from Buell, Binary Quadratic Forms, page 103, the group of binary form classes is isomorphic to the narrow class group of $\mathbb Q ( \sqrt \Delta)$ where $\Delta$ is the discriminant, where I suspect $\Delta$ must be a fundamental discriminant because multiplying by an integer square would not change a field extending $\mathbb Q.$

Then page 103, positive forms we are done, class group and narrow class group are isomorphic. Also done if there is a solution in rational integers to $u^2 - D v^2 = -4.$

Finally the problem: if there is no solution to $u^2 - D v^2 = -4,$ Buell says the class group is the squares of the narrow class group. Buchmann and Vollmer say, page 186, say the class group is a quotient of the narrow class group.

Let's see., examples. I put Positive primes represented by indefinite binary quadratic form with this in mind. Cohen says that $\mathbb Q(\sqrt {205})$ has class number 2. There are four classes of indefinite binary forms of discriminant 205, and $u^2 - 205 y^2 = -4$ is impossible. So, we went from 4 to 2...

In the paper with Pete Clark, he deliberately made no distinction between indefinite form $f$ and the form $-f.$ So, one possibility here is that we are just dividing by 2 to go from 4 to 2..

Maybe this is the quick version: as far as I can tell, if $1$ and $-1$ are distinct as binary forms of discriminant $\Delta,$ the principal genus has even size, call that $E.$ Suppose there are $G$ genera, so that the total number of classes of binary forms of this discriminant is $EG.$ What is the class number of $\mathbb Q ( \sqrt \Delta)?$ So, question, are the numbers the same for positive forms and when the principal form also represents $-1,$ but if indefinite and the principal form does not represent $-1,$ divide by $2?$

Perhaps I can use this to publicize an elementary trick, generally unknown: an indefinite form $\langle a,b,c \rangle$ with positive $\Delta = b^2 - 4 a c$ not a square is reduced, in the sense of Lagrange, Gauss, and Buell, if and only if: $$ac < 0 \; \; \; \; \mbox{and} \; \; \; \; b > |a + c|$$

So, general Question: how to take the class number of binary forms of discriminant $\Delta,$ where either $\Delta \equiv 1 \pmod 4$ is squarefree, or $\Delta \equiv 0 \pmod 4$ and $\Delta/4 \equiv 2,3 \pmod 4$ and this time $\Delta/4$ is squarefree.

To repeat some examples (I've got programs out the wazoo)

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 5 Sun Jun 22 20:03:44 PDT 2014 5 factored 5 1. 1 1 -1 cycle length 2 2. -1 1 1 cycle length 2 5 factored 5 1. 1 1 -1 cycle length 2 form class number is 1 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 12 Sun Jun 22 20:03:52 PDT 2014 12 factored 2^2 * 3 1. 1 2 -2 cycle length 2 2. -1 2 2 cycle length 2 3. 2 2 -1 cycle length 2 4. -2 2 1 cycle length 2 12 factored 2^2 * 3 1. 1 2 -2 cycle length 2 2. -1 2 2 cycle length 2 form class number is 2 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 85 Sun Jun 22 20:04:08 PDT 2014 85 factored 5 * 17 1. 1 9 -1 cycle length 2 2. -1 9 1 cycle length 2 3. 3 7 -3 cycle length 6 4. -3 7 3 cycle length 6 5. 3 5 -5 cycle length 6 6. -3 5 5 cycle length 6 7. 5 5 -3 cycle length 6 8. -5 5 3 cycle length 6 85 factored 5 * 17 1. 1 9 -1 cycle length 2 2. 3 7 -3 cycle length 6 form class number is 2 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 136 Sun Jun 22 20:04:24 PDT 2014 136 factored 2^3 * 17 1. 1 10 -9 cycle length 4 2. -1 10 9 cycle length 4 3. 3 10 -3 cycle length 6 4. -3 10 3 cycle length 6 5. 9 10 -1 cycle length 4 6. -9 10 1 cycle length 4 7. 2 8 -9 cycle length 4 8. -2 8 9 cycle length 4 9. 3 8 -6 cycle length 6 10. -3 8 6 cycle length 6 11. 6 8 -3 cycle length 6 12. -6 8 3 cycle length 6 13. 9 8 -2 cycle length 4 14. -9 8 2 cycle length 4 15. 5 6 -5 cycle length 6 16. -5 6 5 cycle length 6 17. 5 4 -6 cycle length 6 18. -5 4 6 cycle length 6 19. 6 4 -5 cycle length 6 20. -6 4 5 cycle length 6 136 factored 2^3 * 17 1. 1 10 -9 cycle length 4 2. -1 10 9 cycle length 4 3. 3 10 -3 cycle length 6 4. -3 10 3 cycle length 6 form class number is 4 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 205 Sun Jun 22 20:04:32 PDT 2014 205 factored 5 * 41 1. 1 13 -9 cycle length 4 2. -1 13 9 cycle length 4 3. 3 13 -3 cycle length 4 4. -3 13 3 cycle length 4 5. 9 13 -1 cycle length 4 6. -9 13 1 cycle length 4 7. 3 11 -7 cycle length 4 8. -3 11 7 cycle length 4 9. 7 11 -3 cycle length 4 10. -7 11 3 cycle length 4 11. 5 5 -9 cycle length 4 12. -5 5 9 cycle length 4 13. 9 5 -5 cycle length 4 14. -9 5 5 cycle length 4 15. 7 3 -7 cycle length 4 16. -7 3 7 cycle length 4 205 factored 5 * 41 1. 1 13 -9 cycle length 4 2. -1 13 9 cycle length 4 3. 3 13 -3 cycle length 4 4. -3 13 3 cycle length 4 form class number is 4 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle_All_Reduced 221 Sun Jun 22 20:04:40 PDT 2014 221 factored 13 * 17 1. 1 13 -13 cycle length 2 2. -1 13 13 cycle length 2 3. 13 13 -1 cycle length 2 4. -13 13 1 cycle length 2 5. 5 11 -5 cycle length 4 6. -5 11 5 cycle length 4 7. 5 9 -7 cycle length 4 8. -5 9 7 cycle length 4 9. 7 9 -5 cycle length 4 10. -7 9 5 cycle length 4 11. 7 5 -7 cycle length 4 12. -7 5 7 cycle length 4 221 factored 13 * 17 1. 1 13 -13 cycle length 2 2. -1 13 13 cycle length 2 3. 5 11 -5 cycle length 4 4. -5 11 5 cycle length 4 form class number is 4 jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$


in brief, 210 * 4 = 840, 24 reduced forms but 8 SL2 classes,

840    factored   2^3 * 3 * 5 *  7

1.             1          28         -14   cycle length             2
2.            -1          28          14   cycle length             2
3.             2          28          -7   cycle length             2
4.            -2          28           7   cycle length             2
5.             3          24         -22   cycle length             4
6.            -3          24          22   cycle length             4
7.             6          24         -11   cycle length             4
8.            -6          24          11   cycle length             4

form class number is   8