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Properties

Label 417450fd
Number of curves $2$
Conductor $417450$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 417450fd have rank \(2\).

Complex multiplication

The elliptic curves in class 417450fd do not have complex multiplication.

Modular form 417450.2.a.fd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} - q^{12} - 7 q^{13} - q^{14} + q^{16} - 3 q^{17} + q^{18} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().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.

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

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 417450fd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
417450.fd2 417450fd1 \([1, 1, 1, -384238, -48672469]\) \(226646274673/94431744\) \(2613931169256000000\) \([]\) \(11197440\) \(2.2311\) \(\Gamma_0(N)\)-optimal*
417450.fd1 417450fd2 \([1, 1, 1, -14541238, 21334931531]\) \(12284337086925553/1165987944\) \(32275293250946625000\) \([]\) \(33592320\) \(2.7804\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 417450fd1.