Properties

Label 102a
Number of curves 2
Conductor 102
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("102.a1")
sage: E.isogeny_class()

Elliptic curves in class 102a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
102.a1 102a1 [1, 1, 0, -2, 0] 2 8 \(\Gamma_0(N)\)-optimal
102.a2 102a2 [1, 1, 0, 8, 10] 2 16  

Rank

sage: E.rank()

The elliptic curves in class 102a have rank \(1\).

Modular form 102.2.a.a

sage: E.q_eigenform(10)
\( q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 4q^{10} - q^{12} - 6q^{13} + 2q^{14} + 4q^{15} + q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona 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 Cremona labels.