Properties

Label 52371.a
Number of curves 4
Conductor 52371
CM no
Rank 2
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("52371.a1")
sage: E.isogeny_class()

Elliptic curves in class 52371.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
52371.a1 52371e4 [1, -1, 1, -697586, 224210310] 2 540672  
52371.a2 52371e2 [1, -1, 1, -54851, 1566906] 4 270336  
52371.a3 52371e1 [1, -1, 1, -31046, -2080020] 2 135168 \(\Gamma_0(N)\)-optimal
52371.a4 52371e3 [1, -1, 1, 207004, 12041106] 2 540672  

Rank

sage: E.rank()

The elliptic curves in class 52371.a have rank \(2\).

Modular form 52371.2.a.a

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

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 LMFDB 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 LMFDB labels.