Elliptic curve isogeny class with Cremona label 493680e (LMFDB label 493680.e)

• Label: 493680e
• Number of curves: $4$
• Conductor: $493680$
• CM: no
• Rank: $1$

Elliptic curves in class 493680e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
493680.e4	493680e1	\([0, -1, 0, -2448596, 44768747520]\)	\(-3579968623693264/1906997690433375\)	\(-864860860278231103584000\)	\([2]\)	\(77414400\)	\(3.2722\)	\(\Gamma_0(N)\)-optimal^*
493680.e3	493680e2	\([0, -1, 0, -204569416, 1114553823616]\)	\(521902963282042184836/6241849278890625\)	\(11323204352369412624000000\)	\([2, 2]\)	\(154828800\)	\(3.6187\)	\(\Gamma_0(N)\)-optimal^*
493680.e1	493680e3	\([0, -1, 0, -3263657536, 71764805772640]\)	\(1059623036730633329075378/154307373046875\)	\(559851364561500000000000\)	\([2]\)	\(309657600\)	\(3.9653\)	\(\Gamma_0(N)\)-optimal^*
493680.e2	493680e4	\([0, -1, 0, -379414416, -1072687188384]\)	\(1664865424893526702418/826424127435466125\)	\(2998396423421341294328064000\)	\([2]\)	\(309657600\)	\(3.9653\)

^*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 493680e1.

Rank

The elliptic curves in class 493680e have rank \(1\).

Complex multiplication

The elliptic curves in class 493680e do not have complex multiplication.

Modular form 493680.2.a.e

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.