Properties

Label 525a
Number of curves 4
Conductor 525
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("525.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 525a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
525.a3 525a1 [1, 1, 1, -63, 156] [4] 96 \(\Gamma_0(N)\)-optimal
525.a2 525a2 [1, 1, 1, -188, -844] [2, 2] 192  
525.a1 525a3 [1, 1, 1, -2813, -58594] [2] 384  
525.a4 525a4 [1, 1, 1, 437, -4594] [2] 384  

Rank

sage: E.rank()
 

The elliptic curves in class 525a have rank \(1\).

Modular form 525.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{12} + 6q^{13} + q^{14} - q^{16} - 2q^{17} - q^{18} - 8q^{19} + O(q^{20}) \)

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 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.