Properties

Label 128.a
Number of curves 2
Conductor 128
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("128.a1")
sage: E.isogeny_class()

Elliptic curves in class 128.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
128.a1 128a2 [0, 1, 0, -9, 7] 2 8  
128.a2 128a1 [0, 1, 0, 1, 1] 2 4 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

The elliptic curves in class 128.a have rank \(1\).

Modular form 128.2.a.a

sage: E.q_eigenform(10)
\( q - 2q^{3} - 2q^{5} - 4q^{7} + q^{9} + 2q^{11} - 2q^{13} + 4q^{15} - 2q^{17} - 2q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.