Properties

Label 663.b
Number of curves $2$
Conductor $663$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 663.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
663.b1 663c2 \([1, 0, 0, -98, 279]\) \(104154702625/24649677\) \(24649677\) \([2]\) \(128\) \(0.12990\)  
663.b2 663c1 \([1, 0, 0, -33, -72]\) \(3981876625/232713\) \(232713\) \([2]\) \(64\) \(-0.21668\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 663.b do not have complex multiplication.

Modular form 663.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{6} - 2q^{7} + 3q^{8} + q^{9} - 2q^{11} - q^{12} - q^{13} + 2q^{14} - q^{16} + q^{17} - q^{18} + O(q^{20})\)  Toggle raw display

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.