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Properties

Label 469440i
Number of curves $4$
Conductor $469440$
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, 0, 0, -1516908, -682688432]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -1516908, -682688432]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -1516908, -682688432]) 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 469440i have rank \(1\).

Complex multiplication

The elliptic curves in class 469440i do not have complex multiplication.

Modular form 469440.2.a.i

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);
 
\(q - q^{5} - 4 q^{7} + 4 q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 469440i

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
469440.i3 469440i1 \([0, 0, 0, -1516908, -682688432]\) \(2019919152635929/115369574400\) \(22047469007693414400\) \([2]\) \(11796480\) \(2.4659\) \(\Gamma_0(N)\)-optimal*
469440.i2 469440i2 \([0, 0, 0, -4466028, 2777219152]\) \(51549040566902809/12396032640000\) \(2368918728097136640000\) \([2, 2]\) \(23592960\) \(2.8125\) \(\Gamma_0(N)\)-optimal*
469440.i1 469440i3 \([0, 0, 0, -66674028, 209531727952]\) \(171524570744011574809/15247694037600\) \(2913879707722815897600\) \([2]\) \(47185920\) \(3.1591\) \(\Gamma_0(N)\)-optimal*
469440.i4 469440i4 \([0, 0, 0, 10556052, 17456795728]\) \(680707952920628711/1082811037500000\) \(-206928411711897600000000\) \([2]\) \(47185920\) \(3.1591\)  
*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 469440i1.