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Properties

Label 462462.r
Number of curves $4$
Conductor $462462$
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, 1, 0, -2472516, 1495371354]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -2472516, 1495371354]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -2472516, 1495371354]) 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 462462.r have rank \(1\).

Complex multiplication

The elliptic curves in class 462462.r do not have complex multiplication.

Modular form 462462.2.a.r

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

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

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 462462.r

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
462462.r1 462462r3 \([1, 1, 0, -2472516, 1495371354]\) \(8020417344913/187278\) \(39032926498307742\) \([2]\) \(11796480\) \(2.2959\) \(\Gamma_0(N)\)-optimal*
462462.r2 462462r2 \([1, 1, 0, -160206, 21504960]\) \(2181825073/298116\) \(62134046262612324\) \([2, 2]\) \(5898240\) \(1.9493\) \(\Gamma_0(N)\)-optimal*
462462.r3 462462r1 \([1, 1, 0, -41626, -2946236]\) \(38272753/4368\) \(910388956228752\) \([2]\) \(2949120\) \(1.6027\) \(\Gamma_0(N)\)-optimal*
462462.r4 462462r4 \([1, 1, 0, 254824, 114886710]\) \(8780064047/32388174\) \(-6750420311816667486\) \([2]\) \(11796480\) \(2.2959\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 462462.r1.