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Properties

Label 209814.cm
Number of curves $2$
Conductor $209814$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cm1")
 
E.isogeny_class()
 

Elliptic curves in class 209814.cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.cm1 209814bg2 \([1, 1, 1, -42890207, 62415434621]\) \(204055591784617/78708537864\) \(3365669630483050915413576\) \([2]\) \(46448640\) \(3.4053\)  
209814.cm2 209814bg1 \([1, 1, 1, -19111287, -31473253107]\) \(18052771191337/444958272\) \(19026938926121084278848\) \([2]\) \(23224320\) \(3.0587\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 209814.cm have rank \(1\).

Complex multiplication

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

Modular form 209814.2.a.cm

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.