Elliptic curve isogeny class with LMFDB label 422331.bw (Cremona label 422331bw)

• Label: 422331.bw
• Number of curves: $4$
• Conductor: $422331$
• CM: no
• Rank: $1$

Elliptic curves in class 422331.bw

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
422331.bw1	422331bw3	\([1, 0, 1, -204971485, 1129488245771]\)	\(1677087406638588673/4641\)	\(2635481198722281\)	\([2]\)	\(41287680\)	\(3.0774\)	\(\Gamma_0(N)\)-optimal^*
422331.bw2	422331bw2	\([1, 0, 1, -12810880, 17646985241]\)	\(409460675852593/21538881\)	\(12231268243270106121\)	\([2, 2]\)	\(20643840\)	\(2.7308\)	\(\Gamma_0(N)\)-optimal^*
422331.bw3	422331bw4	\([1, 0, 1, -12106995, 19672203163]\)	\(-345608484635233/94427721297\)	\(-53622599464863396417177\)	\([2]\)	\(41287680\)	\(3.0774\)
422331.bw4	422331bw1	\([1, 0, 1, -844835, 243569393]\)	\(117433042273/22801233\)	\(12948119129322566553\)	\([2]\)	\(10321920\)	\(2.3842\)	\(\Gamma_0(N)\)-optimal^*

^*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 422331.bw1.

Rank

The elliptic curves in class 422331.bw have rank \(1\).

Complex multiplication

The elliptic curves in class 422331.bw do not have complex multiplication.

Modular form 422331.2.a.bw

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 LMFDB 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 LMFDB labels.