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

• Label: 473200em
• Number of curves: $4$
• Conductor: $473200$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 473200em have rank \(1\).

Complex multiplication

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

Modular form 473200.2.a.em

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.

Elliptic curves in class 473200em

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
473200.em3	473200em1	\([0, 0, 0, -4508075, -3683819750]\)	\(32798729601/3185\)	\(983896746560000000\)	\([2]\)	\(8257536\)	\(2.4883\)	\(\Gamma_0(N)\)-optimal^*
473200.em2	473200em2	\([0, 0, 0, -4846075, -3099417750]\)	\(40743095121/10144225\)	\(3133711137793600000000\)	\([2, 2]\)	\(16515072\)	\(2.8349\)	\(\Gamma_0(N)\)-optimal^*
473200.em1	473200em3	\([0, 0, 0, -26816075, 50880872250]\)	\(6903498885921/374712065\)	\(115754468336037440000000\)	\([4]\)	\(33030144\)	\(3.1815\)	\(\Gamma_0(N)\)-optimal^*
473200.em4	473200em4	\([0, 0, 0, 11715925, -19677979750]\)	\(575722725759/874680625\)	\(-270202644024040000000000\)	\([2]\)	\(33030144\)	\(3.1815\)

^*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 473200em1.