Properties

Label 63.a
Number of curves 6
Conductor 63
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("63.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 63.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
63.a1 63a5 [1, -1, 0, -7056, 229905] [4] 32  
63.a2 63a4 [1, -1, 0, -441, 3672] [2, 2] 16  
63.a3 63a3 [1, -1, 0, -351, -2430] [2] 16  
63.a4 63a6 [1, -1, 0, -306, 5859] [2] 32  
63.a5 63a2 [1, -1, 0, -36, 27] [2, 2] 8  
63.a6 63a1 [1, -1, 0, 9, 0] [2] 4 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 63.a have rank \(0\).

Modular form 63.2.a.a

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} + 2q^{5} - q^{7} - 3q^{8} + 2q^{10} - 4q^{11} - 2q^{13} - q^{14} - q^{16} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.