Properties

Label 4998.be
Number of curves 6
Conductor 4998
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4998.be1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4998.be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4998.be1 4998bd5 [1, 1, 1, -1359457, 609526973] [2] 49152  
4998.be2 4998bd3 [1, 1, 1, -84967, 9497081] [2, 2] 24576  
4998.be3 4998bd6 [1, 1, 1, -80557, 10532549] [2] 49152  
4998.be4 4998bd2 [1, 1, 1, -5587, 130241] [2, 2] 12288  
4998.be5 4998bd1 [1, 1, 1, -1667, -24991] [2] 6144 \(\Gamma_0(N)\)-optimal
4998.be6 4998bd4 [1, 1, 1, 11073, 776649] [2] 24576  

Rank

sage: E.rank()
 

The elliptic curves in class 4998.be have rank \(1\).

Modular form 4998.2.a.be

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + 2q^{10} - 4q^{11} - q^{12} + 2q^{13} - 2q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.