Properties

Label 209814db
Number of curves 6
Conductor 209814
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("209814.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 209814db

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
209814.t5 209814db1 [1, 1, 0, -1189674, -463696812] [2] 5898240 \(\Gamma_0(N)\)-optimal
209814.t4 209814db2 [1, 1, 0, -3987194, 2525733060] [2, 2] 11796480  
209814.t2 209814db3 [1, 1, 0, -60636974, 181708987200] [2, 2] 23592960  
209814.t6 209814db4 [1, 1, 0, 7902266, 14721941128] [2] 23592960  
209814.t1 209814db5 [1, 1, 0, -970180664, 11630863048182] [2] 47185920  
209814.t3 209814db6 [1, 1, 0, -57489764, 201416186778] [2] 47185920  

Rank

sage: E.rank()
 

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

Modular form 209814.2.a.t

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

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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.