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Properties

Label 417600.fb
Number of curves $2$
Conductor $417600$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, 0, 0, -628500, -191781250]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 417600.fb have rank \(1\).

Complex multiplication

The elliptic curves in class 417600.fb do not have complex multiplication.

Modular form 417600.2.a.fb

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{7} + 3 q^{11} + 4 q^{13} + 8 q^{17} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 417600.fb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
417600.fb1 417600fb1 \([0, 0, 0, -628500, -191781250]\) \(-301302001664/87\) \(-7927875000000\) \([]\) \(2918400\) \(1.8419\) \(\Gamma_0(N)\)-optimal*
417600.fb2 417600fb2 \([0, 0, 0, 1036500, -941481250]\) \(1351431663616/4984209207\) \(-454186063987875000000\) \([]\) \(14592000\) \(2.6466\) \(\Gamma_0(N)\)-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 417600.fb1.