Properties

Label 98.a
Number of curves 6
Conductor 98
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("98.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 98.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
98.a1 98a6 [1, 1, 0, -133795, 18781197] [2] 288  
98.a2 98a5 [1, 1, 0, -8355, 291341] [2] 144  
98.a3 98a4 [1, 1, 0, -1740, 22184] [2] 96  
98.a4 98a2 [1, 1, 0, -515, -4717] [2] 32  
98.a5 98a1 [1, 1, 0, -25, -111] [2] 16 \(\Gamma_0(N)\)-optimal
98.a6 98a3 [1, 1, 0, 220, 2192] [2] 48  

Rank

sage: E.rank()
 

The elliptic curves in class 98.a have rank \(0\).

Modular form 98.2.a.a

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.