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Properties

Label 414736cb
Number of curves $2$
Conductor $414736$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -68232712, -216915221760]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -68232712, -216915221760]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -68232712, -216915221760]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 414736cb have rank \(1\).

Complex multiplication

The elliptic curves in class 414736cb do not have complex multiplication.

Modular form 414736.2.a.cb

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + 2 q^{3} + 2 q^{5} + q^{9} + 2 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 414736cb

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.cb2 414736cb1 \([0, -1, 0, -68232712, -216915221760]\) \(1969910093092/7889\) \(140694515703641424896\) \([2]\) \(24330240\) \(3.0775\) \(\Gamma_0(N)\)-optimal*
414736.cb1 414736cb2 \([0, -1, 0, -69269552, -209981665312]\) \(1030541881826/62236321\) \(2219878068772054402009088\) \([2]\) \(48660480\) \(3.4241\) \(\Gamma_0(N)\)-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 414736cb1.