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Properties

Label 474075.cq
Number of curves $2$
Conductor $474075$
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, -1531253292, 23063535103741]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -1531253292, 23063535103741]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -1531253292, 23063535103741]) 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 474075.cq have rank \(1\).

Complex multiplication

The elliptic curves in class 474075.cq do not have complex multiplication.

Modular form 474075.2.a.cq

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);
 
\(q + q^{2} - q^{4} - 3 q^{8} - 4 q^{11} + 2 q^{13} - q^{16} - 6 q^{17} + 6 q^{19} + O(q^{20})\) 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 474075.cq

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
474075.cq1 474075cq2 \([1, -1, 0, -1531253292, 23063535103741]\) \(8000051600110940079507/144453125\) \(7169666781005859375\) \([2]\) \(123863040\) \(3.6089\) \(\Gamma_0(N)\)-optimal*
474075.cq2 474075cq1 \([1, -1, 0, -95706417, 360361275616]\) \(1953326569433829507/262451171875\) \(13026284122467041015625\) \([2]\) \(61931520\) \(3.2623\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 474075.cq1.