Properties

Label 37.b
Number of curves $3$
Conductor $37$
CM no
Rank $0$
Graph

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

Elliptic curves in class 37.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37.b1 37b2 \([0, 1, 1, -1873, -31833]\) \(727057727488000/37\) \(37\) \([]\) \(6\) \(0.22208\)  
37.b2 37b1 \([0, 1, 1, -23, -50]\) \(1404928000/50653\) \(50653\) \([3]\) \(2\) \(-0.32722\) \(\Gamma_0(N)\)-optimal
37.b3 37b3 \([0, 1, 1, -3, 1]\) \(4096000/37\) \(37\) \([3]\) \(6\) \(-0.87653\)  

Rank

sage: E.rank()
 

The elliptic curves in class 37.b have rank \(0\).

Complex multiplication

The elliptic curves in class 37.b do not have complex multiplication.

Modular form 37.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{4} - q^{7} - 2 q^{9} + 3 q^{11} - 2 q^{12} - 4 q^{13} + 4 q^{16} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.