Properties

Label 91.b
Number of curves $3$
Conductor $91$
CM no
Rank $1$
Graph

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

Elliptic curves in class 91.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
91.b1 91b3 \([0, 1, 1, -117, -1245]\) \(-178643795968/524596891\) \(-524596891\) \([]\) \(36\) \(0.36010\)  
91.b2 91b1 \([0, 1, 1, -7, 5]\) \(-43614208/91\) \(-91\) \([3]\) \(4\) \(-0.73851\) \(\Gamma_0(N)\)-optimal
91.b3 91b2 \([0, 1, 1, 13, 42]\) \(224755712/753571\) \(-753571\) \([3]\) \(12\) \(-0.18920\)  

Rank

sage: E.rank()
 

The elliptic curves in class 91.b have rank \(1\).

Complex multiplication

The elliptic curves in class 91.b do not have complex multiplication.

Modular form 91.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.