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Properties

Label 442225cl
Number of curves $2$
Conductor $442225$
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, 0, 1, -2896, -65537]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -2896, -65537]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -2896, -65537]) 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 442225cl have rank \(0\).

Complex multiplication

The elliptic curves in class 442225cl do not have complex multiplication.

Modular form 442225.2.a.cl

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);
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{8} - 2 q^{9} - q^{12} - 2 q^{13} - q^{16} + 2 q^{17} - 2 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 37 \\ 37 & 1 \end{array}\right)\)

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 442225cl

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
442225.cl2 442225cl1 \([1, 0, 1, -2896, -65537]\) \(-9317\) \(-288156021125\) \([]\) \(314496\) \(0.93725\) \(\Gamma_0(N)\)-optimal*
442225.cl1 442225cl2 \([1, 0, 1, -75117971, 250583824663]\) \(-162677523113838677\) \(-288156021125\) \([]\) \(11636352\) \(2.7427\) \(\Gamma_0(N)\)-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 442225cl1.