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Properties

Label 209814.d
Number of curves $2$
Conductor $209814$
CM no
Rank $2$
Graph

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

Elliptic curves in class 209814.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
209814.d1 209814cl2 [1, 1, 0, -6365086, -6183296390] [2] 6635520  
209814.d2 209814cl1 [1, 1, 0, -420356, -85192356] [2] 3317760 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 209814.d have rank \(2\).

Complex multiplication

The elliptic curves in class 209814.d do not have complex multiplication.

Modular form 209814.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 2q^{10} - q^{12} - 4q^{13} + 2q^{14} + 2q^{15} + q^{16} - q^{18} - 2q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.