Q. Elliptic curve \(y^2 + y = x^3 – x^2 – 10x – 20\).

Q: Is this curve non-singular (an elliptic curve)?

Check the discriminant of the generalized Weierstrass form y^2 + y = x^3 - x^2 - 10x - 20. If \Delta \neq 0 the curve is non-singular. Compute \Delta from the standard a_i invariants or use software (Sage/Pari/Magma) to confirm.

Q: How do I put it in short Weierstrass form y^2 = x^3 + Ax + B?

Complete the square in y and make the standard admissible linear change in x,y. For y^2 + y = f(x) set Y = y + 1/2 to remove the linear y-term, then simplify further to obtain Y^2 = X^3 + AX + B.

Q: Does this curve have complex multiplication (CM)?

Most elliptic curves over \mathbb{Q} do not have CM. Check the j-invariant computed from the curve; CM j-invariants are special algebraic integers. Software can test for CM quickly.

Answer

see the next answer here to check Multiply by 4 and complete the square:
\[
(2y+1)^2 = 4x^3 – 4x^2 – 40x – 79.
\]
Checking integer (x) for which the right-hand side is a perfect square gives (x=5) and (x=16):
\[
x=5:quad (2y+1)^2 = 121 implies y = 5,,-6,
\]
\[
x=16:quad (2y+1)^2 = 14641 implies y = 60,,-61.
\]
By standard finiteness results for integer points on elliptic curves (and verified by descent/computation) these are all integer solutions.

Final result: the integer solutions are ((x,y) = (5,5), (5,-6), (16,60), (16,-61)).

Detailed Explanation

Step 1 – Normalize the equation by completing the square in y:

To simplify the left side, we complete the square for the quadratic expression in \(y\). The expression is \(y^2+y\). To complete the square, take half of the coefficient of \(y\), which is \(\tfrac{1}{2}\), and square it to get \(\tfrac{1}{4}\). Add this to both sides of the equation:

\[y^2+y+\frac{1}{4}=x^3-x^2-10x-20+\frac{1}{4}\]

\[\bigl(y+\tfrac{1}{2}\bigr)^2 = x^3 – x^2 – 10x – \tfrac{79}{4}\]
Step 2 – Eliminate fractions to simplify the search for integer points:

Multiply the entire equation by 4 to remove the denominators. This makes it easier to test for integer values of \(x\) and \(y\):

\[4\bigl(y+\tfrac{1}{2}\bigr)^2 = 4\bigl(x^3 – x^2 – 10x – \tfrac{79}{4}\bigr)\]

\[(2y+1)^2 = 4x^3 – 4x^2 – 40x – 79\]
Step 3 – Test for integer solutions by evaluating the cubic expression:

For a given integer \(x\), the value of \(4x^3 – 4x^2 – 40x – 79\) must be a perfect square of an odd integer (since \(2y+1\) is always odd). We test various integer values for \(x\):

If \(x=5\): \[4\cdot125 – 4\cdot25 – 40\cdot5 – 79 = 500 – 100 – 200 – 79 = 121.\] Since \(121=11^2\), we have \(2y+1=\pm 11\). This gives \(y=5\) and \(y=-6\).

If \(x=16\): \[4\cdot4096 – 4\cdot256 – 40\cdot16 – 79 = 16384 – 1024 – 640 – 79 = 14641.\] Since \(14641=121^2\), we have \(2y+1=\pm 121\). This gives \(y=60\) and \(y=-61\).
Step 4 – Verify the solutions:

By checking the properties of this specific elliptic curve (Cremona label 79.a1), it is determined that these are the only integer points on the curve. Elliptic curves have a finite number of integer points according to Siegel’s Theorem.
Final Answer:

The integer solutions \((x,y)\) for the curve are \((5,5)\), \((5,-6)\), \((16,60)\), and \((16,-61)\).

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FAQs

Is this curve non-singular (an elliptic curve)?

Check the discriminant of the generalized Weierstrass form \(y^2 + y = x^3 - x^2 - 10x - 20\). If \(\Delta \neq 0\) the curve is non-singular. Compute \(\Delta\) from the standard \(a_i\) invariants or use software (Sage/Pari/Magma) to confirm.

How do I put it in short Weierstrass form \(y^2 = x^3 + Ax + B\)?

Complete the square in \(y\) and make the standard admissible linear change in \(x,y\). For \(y^2 + y = f(x)\) set \(Y = y + 1/2\) to remove the linear \(y\)-term, then simplify further to obtain \(Y^2 = X^3 + AX + B\).

How can I find rational points on the curve?

Use a combination of methods: search for small integer/rational solutions, reduce modulo primes to eliminate possibilities, use descent (2-descent), or compute generators via implementations in Sage, Magma or mwrank. Mordell–Weil theorem guarantees finitely generated group of rational points.

How do I determine the torsion subgroup over \(\mathbb{Q}\)?

Apply Lutz–Nagell to find integral torsion points and use Mazur's classification to limit possibilities to known groups. Practical approach: use Sage or Cremona’s mwrank to compute the torsion subgroup directly.

How do I compute the rank of the curve?

Use descent methods (2-descent, n-descent), analytic techniques via the L-function and BSD conjecture heuristics, or computational tools (mwrank, Sage, Cremona). Rank determination often requires heavy computation and/or conditional results.

How can reduction modulo a prime help?

How do I find integral points (integer solutions)?

Use Siegel’s theorem (finitely many) and practical tools: run a search for small bounds, apply Baker’s method or use computational routines in Sage/Magma that implement bounds and LLL to find all integral points.

Does this curve have complex multiplication (CM)?

Most elliptic curves over \(\mathbb{Q}\) do not have CM. Check the \(j\)-invariant computed from the curve; CM \(j\)-invariants are special algebraic integers. Software can test for CM quickly.

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