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Properties

Label 435344e
Number of curves $2$
Conductor $435344$
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, -444864, -114354044]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -444864, -114354044]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -444864, -114354044]) 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 435344e have rank \(2\).

Complex multiplication

The elliptic curves in class 435344e do not have complex multiplication.

Modular form 435344.2.a.e

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);
 
\(q - 2 q^{3} - 2 q^{5} + q^{7} + q^{9} + 2 q^{11} + 4 q^{15} - 2 q^{17} + O(q^{20})\) 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 435344e

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
435344.e2 435344e1 \([0, 1, 0, -444864, -114354044]\) \(1969910093092/7889\) \(38992584909824\) \([2]\) \(2073600\) \(1.8193\) \(\Gamma_0(N)\)-optimal*
435344.e1 435344e2 \([0, 1, 0, -451624, -110706348]\) \(1030541881826/62236321\) \(615225004707203072\) \([2]\) \(4147200\) \(2.1658\) \(\Gamma_0(N)\)-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 435344e1.