Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("1008.e1")
sage: E.isogeny_class()

Elliptic curves in class 1008g

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
1008.e3 1008g1 [0, 0, 0, -66, -205] 2 128 \(\Gamma_0(N)\)-optimal
1008.e2 1008g2 [0, 0, 0, -111, 110] 4 256  
1008.e1 1008g3 [0, 0, 0, -1371, 19514] 2 512  
1008.e4 1008g4 [0, 0, 0, 429, 866] 4 512  

Rank

sage: E.rank()

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

Modular form 1008.2.a.e

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.