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

• Label: 413712ez
• Number of curves: $2$
• Conductor: $413712$
• CM: no
• Rank: $1$

Elliptic curves in class 413712ez

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
413712.ez2	413712ez1	\([0, 0, 0, -17830683, 28932842250]\)	\(43499078731809/82055753\)	\(1182651059908227207168\)	\([2]\)	\(41287680\)	\(2.9335\)	\(\Gamma_0(N)\)-optimal^*
413712.ez1	413712ez2	\([0, 0, 0, -285161643, 1853466644250]\)	\(177930109857804849/634933\)	\(9151146116722741248\)	\([2]\)	\(82575360\)	\(3.2801\)	\(\Gamma_0(N)\)-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 413712ez1.

Rank

The elliptic curves in class 413712ez have rank \(1\).

Complex multiplication

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

Modular form 413712.2.a.ez

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.