Properties

Label 2166.d
Number of curves $4$
Conductor $2166$
CM no
Rank $1$
Graph

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

Elliptic curves in class 2166.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2166.d1 2166d3 \([1, 0, 1, -31606280, 68389818806]\) \(74220219816682217473/16416\) \(772305182496\) \([2]\) \(86400\) \(2.5737\)  
2166.d2 2166d2 \([1, 0, 1, -1975400, 1068459446]\) \(18120364883707393/269485056\) \(12678161875854336\) \([2, 2]\) \(43200\) \(2.2271\)  
2166.d3 2166d4 \([1, 0, 1, -1917640, 1133889974]\) \(-16576888679672833/2216253521952\) \(-104265599459584679712\) \([2]\) \(86400\) \(2.5737\)  
2166.d4 2166d1 \([1, 0, 1, -127080, 15656374]\) \(4824238966273/537919488\) \(25306896220028928\) \([2]\) \(21600\) \(1.8805\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2166.d have rank \(1\).

Complex multiplication

The elliptic curves in class 2166.d do not have complex multiplication.

Modular form 2166.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} + 2q^{15} + q^{16} - 6q^{17} - q^{18} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.