Properties

Label 663.a
Number of curves 6
Conductor 663
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("663.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 663.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
663.a1 663b5 [1, 1, 1, -20174, -1111138] [2] 1024  
663.a2 663b3 [1, 1, 1, -1389, -14094] [2, 2] 512  
663.a3 663b2 [1, 1, 1, -544, 4496] [2, 4] 256  
663.a4 663b1 [1, 1, 1, -539, 4592] [4] 128 \(\Gamma_0(N)\)-optimal
663.a5 663b4 [1, 1, 1, 221, 17042] [4] 512  
663.a6 663b6 [1, 1, 1, 3876, -89910] [2] 1024  

Rank

sage: E.rank()
 

The elliptic curves in class 663.a have rank \(1\).

Modular form 663.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} + 2q^{15} - q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.