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Properties

Label 462462l
Number of curves $2$
Conductor $462462$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -40485491, 103834400685]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -40485491, 103834400685]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -40485491, 103834400685]) 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 462462l have rank \(0\).

Complex multiplication

The elliptic curves in class 462462l do not have complex multiplication.

Modular form 462462.2.a.l

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);
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{8} + q^{9} + 2 q^{10} - q^{12} - q^{13} + 2 q^{15} + q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 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 462462l

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
462462.l2 462462l1 \([1, 1, 0, -40485491, 103834400685]\) \(-46865760129834695603/2632317355746624\) \(-412196730604278208645056\) \([2]\) \(72990720\) \(3.2886\) \(\Gamma_0(N)\)-optimal*
462462.l1 462462l2 \([1, 1, 0, -656260651, 6470580089989]\) \(199611938657816147297843/580879842241608\) \(90960450237204192596952\) \([2]\) \(145981440\) \(3.6352\) \(\Gamma_0(N)\)-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 462462l1.