Properties

Label 1008.d
Number of curves 4
Conductor 1008
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1008.d1")
sage: E.isogeny_class()

Elliptic curves in class 1008.d

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1008.d1 1008h4 [0, 0, 0, -2691, 53730] 2 512  
1008.d2 1008h3 [0, 0, 0, -531, -3726] 2 512  
1008.d3 1008h2 [0, 0, 0, -171, 810] 4 256  
1008.d4 1008h1 [0, 0, 0, 9, 54] 2 128 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

The elliptic curves in class 1008.d have rank \(1\).

Modular form 1008.2.a.d

sage: E.q_eigenform(10)
\( q - 2q^{5} + q^{7} - 4q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.