Properties

Label 138.b
Number of curves 4
Conductor 138
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("138.b1")
sage: E.isogeny_class()

Elliptic curves in class 138.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
138.b1 138b4 [1, 0, 1, -771, 1342] 2 96  
138.b2 138b2 [1, 0, 1, -576, 5266] 6 32  
138.b3 138b1 [1, 0, 1, -36, 82] 6 16 \(\Gamma_0(N)\)-optimal
138.b4 138b3 [1, 0, 1, 189, 190] 2 48  

Rank

sage: E.rank()

The elliptic curves in class 138.b have rank \(0\).

Modular form 138.2.a.b

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

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.