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

Analysis of the Elliptic Curve \( y^2 + y = x^3 – x^2 – 10x – 20 \)

1. Step 1 – Identify the curve equation:

   The given equation is \( y^2 + y = x^3 – x^2 – 10x – 20 \). This is a cubic equation in two variables, which represents an elliptic curve in long Weierstrass form.

2. Step 2 – Determine the Weierstrass coefficients:

   The general long Weierstrass form is \( y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 \). Comparing this to our equation:

   • \( a_1 = 0 \)
   • \( a_2 = -1 \)
   • \( a_3 = 1 \)
   • \( a_4 = -10 \)
   • \( a_6 = -20 \)
3. Step 3 – Identify the curve on LMFDB:

   By searching the database with these coefficients, we identify this as the elliptic curve with the LMFDB label 11.a3 (also known as 11a1 in Cremona’s labeling). It is a very famous curve because it is the modular curve \( X_0(11) \).

4. Step 4 – Determine the Discriminant and Conductor:

   The discriminant \( \Delta \) is -11. Since the discriminant is non-zero, the curve is non-singular. The conductor of this curve is 11, which is the smallest possible conductor for an elliptic curve over the rational numbers \( \mathbb{Q} \).

5. Step 5 – Analyze the Rational Points:

   According to the Mordell-Weil Theorem, the group of rational points \( E(\mathbb{Q}) \) is a finitely generated abelian group. For this specific curve, the rank is 0, meaning there are no points of infinite order. The torsion subgroup is isomorphic to the cyclic group \( \mathbb{Z}/5\mathbb{Z} \).

6. Step 6 – List the Rational Points:

   There are exactly 5 rational points on this curve:

   • The point at infinity \( P_{\infty} \) (the identity element)
   • \( (5, 5) \)
   • \( (5, -6) \)
   • \( (16, 60) \)
   • \( (16, -61) \)
7. Final Summary:

   The curve \( y^2 + y = x^3 – x^2 – 10x – 20 \) is the elliptic curve 11.a3. It has a conductor of 11, a rank of 0, and its rational points form a cyclic group of order 5.

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