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Properties

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

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 209814.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.r1 209814cz1 \([1, 1, 0, -31612704, -67885747200]\) \(81706955619457/744505344\) \(31835923954816977616896\) \([2]\) \(38707200\) \(3.1401\) \(\Gamma_0(N)\)-optimal
209814.r2 209814cz2 \([1, 1, 0, -9232544, -162110696832]\) \(-2035346265217/264305213568\) \(-11302001722115924061635712\) \([2]\) \(77414400\) \(3.4867\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 209814.2.a.r

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