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Properties

Label 209814.ct
Number of curves $2$
Conductor $209814$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ct1")
 
E.isogeny_class()
 

Elliptic curves in class 209814.ct

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.ct1 209814b2 \([1, 0, 0, -1662316440584, 824932833671229504]\) \(2418067440128989194388361/8359273562112\) \(1756163478965154606186251157504\) \([2]\) \(3333980160\) \(5.4472\)  
209814.ct2 209814b1 \([1, 0, 0, -103941139464, 12877489635307584]\) \(591139158854005457801/1097587482427392\) \(230587387442945899866811231371264\) \([2]\) \(1666990080\) \(5.1006\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 209814.ct have rank \(0\).

Complex multiplication

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

Modular form 209814.2.a.ct

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