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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 474320.2.a.gd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - 2 q^{9} - 4 q^{13} - q^{15} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 474320gd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
474320.gd1 474320gd1 \([0, 1, 0, -57631856, -171146074156]\) \(-1693700041/32000\) \(-399967789630494998528000\) \([]\) \(57480192\) \(3.3234\) \(\Gamma_0(N)\)-optimal*
474320.gd2 474320gd2 \([0, 1, 0, 229331744, -800055499916]\) \(106718863559/83886080\) \(-1048491562448964808941240320\) \([]\) \(172440576\) \(3.8727\) \(\Gamma_0(N)\)-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 474320gd1.