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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 91b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
91.b2 | 91b1 | \([0, 1, 1, -7, 5]\) | \(-43614208/91\) | \(-91\) | \([3]\) | \(4\) | \(-0.73851\) | \(\Gamma_0(N)\)-optimal |
91.b3 | 91b2 | \([0, 1, 1, 13, 42]\) | \(224755712/753571\) | \(-753571\) | \([3]\) | \(12\) | \(-0.18920\) | |
91.b1 | 91b3 | \([0, 1, 1, -117, -1245]\) | \(-178643795968/524596891\) | \(-524596891\) | \([]\) | \(36\) | \(0.36010\) |
Rank
sage: E.rank()
The elliptic curves in class 91b have rank \(1\).
Complex multiplication
The elliptic curves in class 91b do not have complex multiplication.Modular form 91.2.a.b
sage: E.q_eigenform(10)
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 Cremona 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 Cremona labels.