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Properties

Label 413205.d
Number of curves $4$
Conductor $413205$
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([1, 0, 0, -1188665, 489013290]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -1188665, 489013290]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -1188665, 489013290]) 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 413205.d have rank \(0\).

Complex multiplication

The elliptic curves in class 413205.d do not have complex multiplication.

Modular form 413205.2.a.d

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);
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + q^{15} - q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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 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 413205.d

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
413205.d1 413205d3 \([1, 0, 0, -1188665, 489013290]\) \(38480618749557529/857682789615\) \(4139871008058788535\) \([2]\) \(8847360\) \(2.3597\) \(\Gamma_0(N)\)-optimal*
413205.d2 413205d2 \([1, 0, 0, -161990, -13852125]\) \(97393143178729/39221822025\) \(189316243546668225\) \([2, 2]\) \(4423680\) \(2.0131\) \(\Gamma_0(N)\)-optimal*
413205.d3 413205d1 \([1, 0, 0, -140865, -20354400]\) \(64043209720729/24755625\) \(119490673550625\) \([2]\) \(2211840\) \(1.6666\) \(\Gamma_0(N)\)-optimal*
413205.d4 413205d4 \([1, 0, 0, 526685, -100487440]\) \(3347467708032071/2841729286815\) \(-13716484497162223335\) \([2]\) \(8847360\) \(2.3597\)  
*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 413205.d1.