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Properties

Label 493680.bb
Number of curves $4$
Conductor $493680$
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([0, -1, 0, -360136, -82891664]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -360136, -82891664]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -360136, -82891664]) 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 493680.bb have rank \(1\).

Complex multiplication

The elliptic curves in class 493680.bb do not have complex multiplication.

Modular form 493680.2.a.bb

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);
 
\(q - q^{3} - q^{5} + q^{9} - 2 q^{13} + q^{15} - q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

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 493680.bb

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
493680.bb1 493680bb4 \([0, -1, 0, -360136, -82891664]\) \(711882749089/1721250\) \(12489930224640000\) \([2]\) \(3932160\) \(1.9675\)  
493680.bb2 493680bb3 \([0, -1, 0, -321416, 69943920]\) \(506071034209/2505630\) \(18181637687009280\) \([2]\) \(3932160\) \(1.9675\) \(\Gamma_0(N)\)-optimal*
493680.bb3 493680bb2 \([0, -1, 0, -31016, -216720]\) \(454756609/260100\) \(1887367233945600\) \([2, 2]\) \(1966080\) \(1.6209\) \(\Gamma_0(N)\)-optimal*
493680.bb4 493680bb1 \([0, -1, 0, 7704, -30864]\) \(6967871/4080\) \(-29605760532480\) \([2]\) \(983040\) \(1.2743\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 493680.bb1.