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Properties

Label 432960.gc
Number of curves $2$
Conductor $432960$
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, -36465, 2670063]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -36465, 2670063]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -36465, 2670063]) 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 432960.gc have rank \(1\).

Complex multiplication

The elliptic curves in class 432960.gc do not have complex multiplication.

Modular form 432960.2.a.gc

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);
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} + q^{11} + 4 q^{13} + q^{15} + 3 q^{17} + 2 q^{19} + O(q^{20})\) 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.

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

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 432960.gc

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
432960.gc1 432960gc2 \([0, 1, 0, -36465, 2670063]\) \(-327298742232784/284299125\) \(-4657956864000\) \([]\) \(912384\) \(1.3570\) \(\Gamma_0(N)\)-optimal*
432960.gc2 432960gc1 \([0, 1, 0, 495, 16335]\) \(817036976/7367085\) \(-120702320640\) \([]\) \(304128\) \(0.80765\) \(\Gamma_0(N)\)-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 432960.gc1.