(Short) Weierstrass forms |
[C1] y^{2} = x^{3} + b [C2] y^{2} = x^{3} + ax [C3] y^{2} = x^{3} + ax + b |
Related to Diophantine equations |
[C4] X^{3} + Y^{3} = A [C5] y^{2} = Dx^{4}+1 [C6] Congruent number D : y^{2} = x^{3} - D^{2}x [C7] Concordant number n : y^{2} = x(x+1)(x+n) [C8] y^{2} = (x+p)(x^{2}+p^{2}) (by Stroeker) [C9] (x+y+z)(1/x+1/y+1/z)=n (by Bremner) [C10] x/y+y/z+z/x=n |
[C1] y^{2} = x^{3} + b (b∈Z)
-10000≦b≦-1 (Aug. 07, 2000)
-1000≦b≦-1 (Sep. 18, 2000) |
This curve is the special case of the short Weierstrass form, y^{2} = x^{3} + ax + b, a=0If b=0, the curve becomes singular (cusp). By Hall's conjecture, if x, y are positive integers such that b=x^{3}-y^{2} is nonzero, then |b| ≫_{ε} x^{1/2 - ε} Structure of torsion group (Aug. 16, 2000) |
[C2] y^{2} = x^{3} + ax (a∈Z)
-10000≦a≦-1 (Aug. 07, 2000)
-1000≦a≦-1 (Sep. 18, 2000) |
This curve is the special case of the short Weierstrass form, y^{2} = x^{3} + ax + b, b=0If a=0, the curve becomes singular (node). Structure of torsion group (Aug. 16, 2000) |
[C3] y^{2} = x^{3} + ax + b (a,b∈Z)
1≦a≦100, 1≦b≦100 (Aug. 16, 2000)
1≦a≦20, 1≦b≦20 (Feb. 01, 2008) |
This curve is called as short Weierstrass form. (0, 0),the curve becomes singular. |
[C4] X^{3} + Y^{3} = A (A∈Z) |
The cubic equation is birationally equivalent to the elliptic curve,
E : y^{2}=x^{3}-423.A^{2}under the following birational transformation. |
[C5] Y^{2}=DX^{4}+1 (D∈Z) |
This curve is birationally equivalent to,
E : y^{2}=x^{3}-4Dxunder the following birational transformation. |
[C6] Congruent Numbers D
1≦D≦999 (Jul. 30, 2002) : 361 1001≦D≦1999 (Jul. 30, 2002) : 358 2001≦D≦2999 (Jul. 30, 2002) : 354 3001≦D≦3999 (Jul. 30, 2002) : 354 4001≦D≦4999 (Jul. 30, 2002) : 349 5001≦D≦5999 (Jul. 30, 2002) : 345 6001≦D≦6999 (Jul. 30, 2002) : 351 7001≦D≦7999 (Jul. 30, 2002) : 340 8001≦D≦8999 (Jul. 30, 2002) : 347 9001≦D≦9999 (Jul. 30, 2002) : 352 |
The definition of congruent number and the birational transformation
between the defined curve and the elliptic curve
E : y^{2}=x(x^{2}-D^{2})is described here. The criteria for the congruent number was given in Tunell's paper, J. B. Tunell, "A Classical Diophantine Problem and Modular Forms of Weight 3/2", Inventiones mathematicae, 72(1983)323-334. |
[C7] Concordant Numbers n |
Definition of concordant number
If there exists the solution of following two quadratic forms simultaneously,
x^{2} + y^{2} = z^{2}n is called as concordant. Historically, it is studied by Euler. If n is concordant, the following elliptic curve, E : y^{2}=x(x+1)(x+n)has a rank larger than 0 (i.e. it has non-trivial rational points). Birational transformation is here. |
[C8] y^{2}=(x+p)(x^{2}+p^{2}) (p∈Z, prime)
1≦n≦1000 (January 31, 2008) |
This curve is studied by,
In the former paper, they showed that when p=2 or p=±3 (mod 8), the rank is 0. In the latter paper, they determined all integer solutions of cases p=167, 223, 337, 1201, and determined the ranks of these elliptic curves are 1, 1, 3, and 3 respectively. |
[C9] (x+y+z)(1/x+1/y+1/z)=n (n, x, y, z∈Z) |
This diophantine equation is birationally equivalent to the elliptic curve, E : y^{2}=x^{3}+(n^{2}-6n-3)x^{2}+16nxbirational transformation is here. |
[C10] x/y+y/z+z/x=n (n, x, y, z∈Z) |
This diophantine equation is birationally equivalent to the elliptic curve, E : y^{2}+nxy=x^{3}birational transformation is here. |
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