Elliptic curve isogeny class with LMFDB label 413712.dy (Cremona label 413712dy)

• Label: 413712.dy
• Number of curves: $2$
• Conductor: $413712$
• CM: no
• Rank: $2$

Elliptic curves in class 413712.dy

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
413712.dy1	413712dy2	\([0, 0, 0, -79599, 2051998]\)	\(61918288/33813\)	\(30458696394417408\)	\([2]\)	\(2752512\)	\(1.8537\)	\(\Gamma_0(N)\)-optimal^*
413712.dy2	413712dy1	\([0, 0, 0, 19266, 252655]\)	\(14047232/8619\)	\(-485248839616944\)	\([2]\)	\(1376256\)	\(1.5071\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 413712.dy1.

Rank

The elliptic curves in class 413712.dy have rank \(2\).

Complex multiplication

The elliptic curves in class 413712.dy do not have complex multiplication.

Modular form 413712.2.a.dy

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.