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Properties

Label 486720.ni
Number of curves $2$
Conductor $486720$
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([0, 0, 0, -2186445612, 37900442887216]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -2186445612, 37900442887216]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -2186445612, 37900442887216]) 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 486720.ni have rank \(0\).

Complex multiplication

The elliptic curves in class 486720.ni do not have complex multiplication.

Modular form 486720.2.a.ni

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);
 
\(q + q^{5} + 6 q^{11} - 6 q^{17} + 6 q^{19} + O(q^{20})\) 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 486720.ni

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
486720.ni1 486720ni1 \([0, 0, 0, -2186445612, 37900442887216]\) \(570403428460237/23887872000\) \(48410000185925514723065856000\) \([2]\) \(517570560\) \(4.2689\) \(\Gamma_0(N)\)-optimal
486720.ni2 486720ni2 \([0, 0, 0, 1053162708, 140610280907824]\) \(63745936931123/4251528000000\) \(-8615939974496997127225344000000\) \([2]\) \(1035141120\) \(4.6155\)