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Properties

Label 486330p
Number of curves $2$
Conductor $486330$
CM no
Rank $2$
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, -34373, 2405756]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -34373, 2405756]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -34373, 2405756]) 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 486330p have rank \(2\).

Complex multiplication

The elliptic curves in class 486330p do not have complex multiplication.

Modular form 486330.2.a.p

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);
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 2 q^{11} + q^{12} - q^{13} + q^{15} + q^{16} - 2 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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 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 486330p

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
486330.p1 486330p1 \([1, 0, 1, -34373, 2405756]\) \(4491153137595264841/92819120062500\) \(92819120062500\) \([2]\) \(1966080\) \(1.4704\) \(\Gamma_0(N)\)-optimal
486330.p2 486330p2 \([1, 0, 1, 1877, 7234256]\) \(731895525155159/22607933869667250\) \(-22607933869667250\) \([2]\) \(3932160\) \(1.8170\)