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Properties

Label 445200.a
Number of curves $2$
Conductor $445200$
CM no
Rank $2$
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, -1033408, 404509312]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -1033408, 404509312]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -1033408, 404509312]) 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 445200.a have rank \(2\).

Complex multiplication

The elliptic curves in class 445200.a do not have complex multiplication.

Modular form 445200.2.a.a

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);
 
\(q - q^{3} - q^{7} + q^{9} - 6 q^{11} - 2 q^{13} - 6 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 445200.a

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
445200.a1 445200a2 \([0, -1, 0, -1033408, 404509312]\) \(1907039182132729/1003402890\) \(64217784960000000\) \([2]\) \(6193152\) \(2.1753\) \(\Gamma_0(N)\)-optimal*
445200.a2 445200a1 \([0, -1, 0, -53408, 8589312]\) \(-263251475929/343583100\) \(-21989318400000000\) \([2]\) \(3096576\) \(1.8288\) \(\Gamma_0(N)\)-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 445200.a1.