Properties

Label 45.a
Number of curves 8
Conductor 45
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("45.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 45.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
45.a1 45a7 [1, -1, 0, -19440, 1048135] [2] 32  
45.a2 45a5 [1, -1, 0, -1215, 16600] [2, 2] 16  
45.a3 45a8 [1, -1, 0, -990, 22765] [2] 32  
45.a4 45a3 [1, -1, 0, -720, -7259] [2] 8  
45.a5 45a4 [1, -1, 0, -90, 175] [2, 2] 8  
45.a6 45a2 [1, -1, 0, -45, -104] [2, 2] 4  
45.a7 45a1 [1, -1, 0, 0, -5] [2] 2 \(\Gamma_0(N)\)-optimal
45.a8 45a6 [1, -1, 0, 315, 1066] [2] 16  

Rank

sage: E.rank()
 

The elliptic curves in class 45.a have rank \(0\).

Modular form 45.2.a.a

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} + 4q^{11} - 2q^{13} - q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.