Properties

Label 225.a
Number of curves 2
Conductor 225
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("225.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 225.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
225.a1 225e1 [0, 0, 1, -75, 256] [] 48 \(\Gamma_0(N)\)-optimal
225.a2 225e2 [0, 0, 1, 375, -12344] [] 240  

Rank

sage: E.rank()
 

The elliptic curves in class 225.a have rank \(1\).

Modular form 225.2.a.a

sage: E.q_eigenform(10)
 
\( q - 2q^{2} + 2q^{4} - 3q^{7} - 2q^{11} + q^{13} + 6q^{14} - 4q^{16} - 2q^{17} - 5q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.