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Properties

Label 474320ip
Number of curves $2$
Conductor $474320$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ip1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 474320ip have rank \(0\).

Complex multiplication

The elliptic curves in class 474320ip do not have complex multiplication.

Modular form 474320.2.a.ip

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} + q^{5} + q^{9} - 6 q^{13} + 2 q^{15} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 474320ip

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
474320.ip2 474320ip1 \([0, -1, 0, -161520, -37154368]\) \(-726572699/512000\) \(-328394749247488000\) \([2]\) \(6635520\) \(2.0611\) \(\Gamma_0(N)\)-optimal*
474320.ip1 474320ip2 \([0, -1, 0, -2921200, -1920360000]\) \(4298149261979/1000000\) \(641395994624000000\) \([2]\) \(13271040\) \(2.4077\) \(\Gamma_0(N)\)-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 474320ip1.