Properties

Label 660.d
Number of curves $4$
Conductor $660$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 660.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
660.d1 660d4 \([0, 1, 0, -249996, -48194796]\) \(6749703004355978704/5671875\) \(1452000000\) \([2]\) \(1728\) \(1.4936\)  
660.d2 660d3 \([0, 1, 0, -15621, -757296]\) \(-26348629355659264/24169921875\) \(-386718750000\) \([2]\) \(864\) \(1.1470\)  
660.d3 660d2 \([0, 1, 0, -3156, -63900]\) \(13584145739344/1195803675\) \(306125740800\) \([6]\) \(576\) \(0.94425\)  
660.d4 660d1 \([0, 1, 0, 219, -4500]\) \(72268906496/606436875\) \(-9702990000\) \([6]\) \(288\) \(0.59767\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 660.d have rank \(0\).

Complex multiplication

The elliptic curves in class 660.d do not have complex multiplication.

Modular form 660.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + 2q^{7} + q^{9} + q^{11} + 2q^{13} - q^{15} + 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.