Elliptic curve isogeny class with Cremona label 458640df (LMFDB label 458640.df)

• Label: 458640df
• Number of curves: $4$
• Conductor: $458640$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 458640df have rank \(0\).

Complex multiplication

The elliptic curves in class 458640df do not have complex multiplication.

Modular form 458640.2.a.df

Isogeny matrix

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 458640df

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
458640.df4	458640df1	\([0, 0, 0, 45717, -3976742]\)	\(30080231/36855\)	\(-12947088955207680\)	\([2]\)	\(2359296\)	\(1.7778\)	\(\Gamma_0(N)\)-optimal^*
458640.df3	458640df2	\([0, 0, 0, -271803, -38205398]\)	\(6321363049/1863225\)	\(654547274957721600\)	\([2, 2]\)	\(4718592\)	\(2.1243\)	\(\Gamma_0(N)\)-optimal^*
458640.df2	458640df3	\([0, 0, 0, -1647723, 784319578]\)	\(1408317602329/58524375\)	\(20559497739056640000\)	\([2]\)	\(9437184\)	\(2.4709\)	\(\Gamma_0(N)\)-optimal^*
458640.df1	458640df4	\([0, 0, 0, -3976203, -3051364358]\)	\(19790357598649/2998905\)	\(1053509423503380480\)	\([2]\)	\(9437184\)	\(2.4709\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 458640df1.