Properties

Label 637.b
Number of curves $3$
Conductor $637$
CM no
Rank $0$
Graph

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

Elliptic curves in class 637.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
637.b1 637b3 \([0, -1, 1, -5749, 415463]\) \(-178643795968/524596891\) \(-61718299629259\) \([]\) \(1728\) \(1.3331\)  
637.b2 637b1 \([0, -1, 1, -359, -2507]\) \(-43614208/91\) \(-10706059\) \([]\) \(192\) \(0.23445\) \(\Gamma_0(N)\)-optimal
637.b3 637b2 \([0, -1, 1, 621, -13238]\) \(224755712/753571\) \(-88656874579\) \([]\) \(576\) \(0.78375\)  

Rank

sage: E.rank()
 

The elliptic curves in class 637.b have rank \(0\).

Complex multiplication

The elliptic curves in class 637.b do not have complex multiplication.

Modular form 637.2.a.b

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} - 2 q^{4} + 3 q^{5} + q^{9} - 4 q^{12} - q^{13} + 6 q^{15} + 4 q^{16} + 6 q^{17} + 7 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.