Elliptic curve isogeny class with LMFDB label 422370.e (Cremona label 422370e)

• Label: 422370.e
• Number of curves: $4$
• Conductor: $422370$
• CM: no
• Rank: $1$

Elliptic curves in class 422370.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
422370.e1	422370e3	\([1, -1, 0, -674235, -204369179]\)	\(988345570681/44994560\)	\(1543153553531965440\)	\([2]\)	\(12317184\)	\(2.2520\)
422370.e2	422370e1	\([1, -1, 0, -105660, 13167616]\)	\(3803721481/26000\)	\(891707628474000\)	\([2]\)	\(4105728\)	\(1.7027\)	\(\Gamma_0(N)\)-optimal^*
422370.e3	422370e2	\([1, -1, 0, -40680, 29139700]\)	\(-217081801/10562500\)	\(-362256224067562500\)	\([2]\)	\(8211456\)	\(2.0492\)
422370.e4	422370e4	\([1, -1, 0, 365445, -778480475]\)	\(157376536199/7722894400\)	\(-264867840399197505600\)	\([2]\)	\(24634368\)	\(2.5985\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 422370.e1.

Rank

The elliptic curves in class 422370.e have rank \(1\).

Complex multiplication

The elliptic curves in class 422370.e do not have complex multiplication.

Modular form 422370.2.a.e

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.