Properties

Label 22743n
Number of curves 6
Conductor 22743
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("22743.f1")
sage: E.isogeny_class()

Elliptic curves in class 22743n

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
22743.f6 22743n1 [1, -1, 1, 3181, -16014] 2 27648 \(\Gamma_0(N)\)-optimal
22743.f5 22743n2 [1, -1, 1, -13064, -119982] 4 55296  
22743.f3 22743n3 [1, -1, 1, -126779, 17301156] 2 110592  
22743.f2 22743n4 [1, -1, 1, -159269, -24390012] 4 110592  
22743.f4 22743n5 [1, -1, 1, -110534, -39634320] 2 221184  
22743.f1 22743n6 [1, -1, 1, -2547284, -1564182084] 2 221184  

Rank

sage: E.rank()

The elliptic curves in class 22743n have rank \(1\).

Modular form 22743.2.a.f

sage: E.q_eigenform(10)
\( q - q^{2} - q^{4} + 2q^{5} - q^{7} + 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} + q^{14} - q^{16} + 6q^{17} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.