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Properties

Label 442225bc
Number of curves $2$
Conductor $442225$
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([1, 0, 0, -3547013, 2799247142]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -3547013, 2799247142]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -3547013, 2799247142]) 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 442225bc have rank \(1\).

Complex multiplication

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

Modular form 442225.2.a.bc

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 442225bc

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.bc2 442225bc1 \([1, 0, 0, -3547013, 2799247142]\) \(-9317\) \(-529707308270861328125\) \([]\) \(11007360\) \(2.7149\) \(\Gamma_0(N)\)-optimal
442225.bc1 442225bc2 \([1, 0, 0, -92019513888, -10744057540978483]\) \(-162677523113838677\) \(-529707308270861328125\) \([]\) \(407272320\) \(4.5204\)