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Properties

Label 450450.jl
Number of curves $2$
Conductor $450450$
CM no
Rank $0$
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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -150530, 22515347]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -150530, 22515347]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -150530, 22515347]) 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 450450.jl have rank \(0\).

Complex multiplication

The elliptic curves in class 450450.jl do not have complex multiplication.

Modular form 450450.2.a.jl

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);
 
\(q + q^{2} + q^{4} + q^{7} + q^{8} - q^{11} - q^{13} + q^{14} + q^{16} + 4 q^{19} + O(q^{20})\) 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 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 450450.jl

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
450450.jl1 450450jl2 \([1, -1, 1, -150530, 22515347]\) \(33116363266897/2576574\) \(29348788218750\) \([2]\) \(2097152\) \(1.6316\) \(\Gamma_0(N)\)-optimal*
450450.jl2 450450jl1 \([1, -1, 1, -8780, 402347]\) \(-6570725617/2270268\) \(-25859771437500\) \([2]\) \(1048576\) \(1.2851\) \(\Gamma_0(N)\)-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 450450.jl1.