Elliptic curve isogeny class with LMFDB label 489762.r (Cremona label 489762r)

• Label: 489762.r
• Number of curves: $4$
• Conductor: $489762$
• CM: no
• Rank: $1$

Elliptic curves in class 489762.r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
489762.r1	489762r4	\([1, -1, 0, -689692158, -6971009505836]\)	\(10311027240444450437737/676744245358272\)	\(2381289591147072809740992\)	\([2]\)	\(222953472\)	\(3.7339\)
489762.r2	489762r2	\([1, -1, 0, -45761598, -94732627820]\)	\(3011893562835369577/640946843357184\)	\(2255327706195735494529024\)	\([2, 2]\)	\(111476736\)	\(3.3874\)
489762.r3	489762r1	\([1, -1, 0, -14611518, 20204937364]\)	\(98044243279969897/6636680773632\)	\(23352779065966253309952\)	\([2]\)	\(55738368\)	\(3.0408\)	\(\Gamma_0(N)\)-optimal^*
489762.r4	489762r3	\([1, -1, 0, 99767682, -574833722540]\)	\(31210858401683537303/58626037175299776\)	\(-206290002542740150104697536\)	\([2]\)	\(222953472\)	\(3.7339\)

^*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 489762.r1.

Rank

The elliptic curves in class 489762.r have rank \(1\).

Complex multiplication

The elliptic curves in class 489762.r do not have complex multiplication.

Modular form 489762.2.a.r

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.