Elliptic curve isogeny class with Cremona label 473200ez (LMFDB label 473200.ez)

• Label: 473200ez
• Number of curves: $3$
• Conductor: $473200$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 473200ez have rank \(0\).

Complex multiplication

The elliptic curves in class 473200ez do not have complex multiplication.

Modular form 473200.2.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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 473200ez

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
473200.ez2	473200ez1	\([0, 1, 0, -39103052408, -3227203597664812]\)	\(-21405018343206000779641/2177246093750000000\)	\(-672585666593750000000000000000000\)	\([]\)	\(2048385024\)	\(5.0389\)	\(\Gamma_0(N)\)-optimal^*
473200.ez3	473200ez2	\([0, 1, 0, 240803197592, 2960213839835188]\)	\(4998853083179567995470359/2905108466204672000000\)	\(-897433836201786025705472000000000000\)	\([]\)	\(6145155072\)	\(5.5882\)	\(\Gamma_0(N)\)-optimal^*
473200.ez1	473200ez3	\([0, 1, 0, -3414016152408, 2574946340740935188]\)	\(-14245586655234650511684983641/1028175397808386133196800\)	\(-317619600878086301644128832716800000000\)	\([]\)	\(18435465216\)	\(6.1375\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 473200ez1.