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

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

Elliptic curves in class 422370cw

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
422370.cw4	422370cw1	\([1, -1, 0, -937404, 577349968]\)	\(-2656166199049/2658140160\)	\(-91164763777888419840\)	\([2]\)	\(17694720\)	\(2.5256\)	\(\Gamma_0(N)\)-optimal^*
422370.cw3	422370cw2	\([1, -1, 0, -17572284, 28347618640]\)	\(17496824387403529/6580454400\)	\(225686207204049945600\)	\([2, 2]\)	\(35389440\)	\(2.8722\)	\(\Gamma_0(N)\)-optimal^*
422370.cw1	422370cw3	\([1, -1, 0, -281131164, 1814380724848]\)	\(71647584155243142409/10140000\)	\(347765975104860000\)	\([2]\)	\(70778880\)	\(3.2188\)	\(\Gamma_0(N)\)-optimal^*
422370.cw2	422370cw4	\([1, -1, 0, -20171484, 19412088880]\)	\(26465989780414729/10571870144160\)	\(362577586722461465415840\)	\([2]\)	\(70778880\)	\(3.2188\)

^*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 422370cw1.

Rank

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

Complex multiplication

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

Modular form 422370.2.a.cw

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.