Properties

Label 1005.a
Number of curves 2
Conductor 1005
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1005.a1")
sage: E.isogeny_class()

Elliptic curves in class 1005.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1005.a1 1005b2 [0, 1, 1, -3001, -70904] 1 1440  
1005.a2 1005b1 [0, 1, 1, 239, 295] 3 480 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

The elliptic curves in class 1005.a have rank \(1\).

Modular form 1005.2.a.a

sage: E.q_eigenform(10)
\( q + q^{3} - 2q^{4} - q^{5} + 2q^{7} + q^{9} - 6q^{11} - 2q^{12} + 2q^{13} - q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.