Properties

Label 138.a
Number of curves 2
Conductor 138
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("138.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 138.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
138.a1 138a2 [1, 1, 0, -31, 55] [2] 16  
138.a2 138a1 [1, 1, 0, -1, 1] [2] 8 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 138.a have rank \(1\).

Modular form 138.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 2q^{10} - 6q^{11} - q^{12} - 2q^{13} + 2q^{14} + 2q^{15} + q^{16} - q^{18} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.