[C1]  y² = x³ + b

[C2]  y² = x³ + ax

[C3]  y² = x³ + ax + b

[C4]  X³ + Y³ = A

[C5]  y² = Dx⁴+1

[C6]  Congruent number D : y² = x³ - D²x

[C7]  Concordant number n : y² = x(x+1)(x+n)

[C8]  y² = (x+p)(x²+p²)    (by Stroeker)

[C9]  (x+y+z)(1/x+1/y+1/z)=n    (by Bremner)

[C10]  x/y+y/z+z/x=n

• integer point (x≦10⁹)

-10000≦b≦-1  (Aug. 07, 2000)
 1≦b≦10000  (Aug. 07, 2000)

• rank / generator

-1000≦b≦-1  (Sep. 18, 2000)
 1≦b≦1000  (Sep. 18, 2000)

This curve is the special case of the short Weierstrass form,

y² = x³ + ax + b, a=0

By Hall's conjecture, if x, y are positive integers such that b=x³-y² is nonzero, then

|b| ≫_ε x^{1/2 - ε}

Structure of torsion group (Aug. 16, 2000)

• integer point (x≦10⁹)

-10000≦a≦-1  (Aug. 07, 2000)
 1≦a≦10000  (Aug. 07, 2000)

• rank / generator

-1000≦a≦-1  (Sep. 18, 2000)
 1≦a≦1000  (Sep. 18, 2000)

This curve is the special case of the short Weierstrass form,

y² = x³ + ax + b, b=0

Structure of torsion group (Aug. 16, 2000)

• integer point (x≦10⁹)

 1≦a≦100, 1≦b≦100 (Aug. 16, 2000)
 1≦a≦100, -100≦b≦-1 (Aug. 16, 2000)
-100≦a≦-1, 1≦b≦100 (Aug. 16, 2000)
-100≦a≦-1, -100≦b≦-1 (Aug. 16, 2000)

• rank / generator

 1≦a≦20, 1≦b≦20 (Feb. 01, 2008)
 1≦a≦20, -20≦b≦-1 (Feb. 01, 2008)
-20≦a≦-1, 1≦b≦20 (Feb. 01, 2008)
-20≦a≦-1, -20≦b≦-1 (Feb. 01, 2008)

This curve is called as short Weierstrass form.
If the discriminant D=a²-4b=0,
i.e. (a, b) =

(0, 0),
(±2, 1), (±4, 4), (±6, 9),
(±8, 16), (±10, 25), (±12, 36),
(±14, 49), (±16, 64), (±18, 81),
(±20, 100), ...

• rank / generator of y²=x³-423.A²

1≦A≦1000 (January 31, 2008)


• rational solutions of X³ + Y³ = A

1≦A≦1000 (January 31, 2008)

E : y²=x³-423.A²
  (A ∈Z; X, Y ∈Q)

• rank / generator of y²=x³-4Dx

1≦D≦1000 (January 31, 2008)


• rational solutions of Y²=DX⁴+1

1≦D≦1000 (February 05, 2008)

E : y²=x³-4Dx

• rank / generator of y²=x(x²-D²)

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

E : y²=x(x²-D²)

J. B. Tunell, "A Classical Diophantine Problem and Modular Forms of Weight 3/2", Inventiones mathematicae, 72(1983)323-334.

• rank / generator of y²=x(x+1)(x+n)

1≦n≦1000 (February 01, 2008)


• representations of Concordant numbers

1≦n≦1000 (February 05, 2008)

x² + y² = z²
x² + ny² = w²

E : y²=x(x+1)(x+n)

• rank / generator

1≦n≦1000 (January 31, 2008)

• R. J. Stroeker and J. Top,
"On the equation Y²=(X+p)(X²+p²)", Rocky Mountain J. Math, 24:3 (1994), 1135-1161
• Benjamin M. M. de Weger,
"Solving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms", Experimental Mathematics, Vol. 7 (1998), No. 3, page 243-256.

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.

• rank / generator of
E : y²=x³+(n²-6n-3)x²+16nx

-500≦n≦-1 (February 05, 2008)
1≦n≦500 (February 05, 2008)


• integer solution of
(x+y+z)(1/x+1/y+1/z)=n

-500≦n≦-1 (February 05, 2008)
1≦n≦500 (February 05, 2008)

This diophantine equation is birationally equivalent to the elliptic curve,

E : y²=x³+(n²-6n-3)x²+16nx

• rank / generator of
E : y²+nxy=x³

-100≦n≦-1 (March 07, 2008)
1≦n≦100 (March 07, 2008)


• integer solution of
x/y+y/z+z/x=n

-100≦n≦-1 (March 07, 2008)
1≦n≦100 (March 07, 2008)

This diophantine equation is birationally equivalent to the elliptic curve,

E : y²+nxy=x³