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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 87451d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
87451.d2 | 87451d1 | \([0, -1, 1, -7047, -225773]\) | \(-43614208/91\) | \(-80762834971\) | \([]\) | \(119880\) | \(0.97849\) | \(\Gamma_0(N)\)-optimal |
87451.d3 | 87451d2 | \([0, -1, 1, 12173, -1135840]\) | \(224755712/753571\) | \(-668797036394851\) | \([]\) | \(359640\) | \(1.5278\) | |
87451.d1 | 87451d3 | \([0, -1, 1, -112757, 35955877]\) | \(-178643795968/524596891\) | \(-465581671803655771\) | \([]\) | \(1078920\) | \(2.0771\) |
Rank
sage: E.rank()
The elliptic curves in class 87451d have rank \(1\).
Complex multiplication
The elliptic curves in class 87451d do not have complex multiplication.Modular form 87451.2.a.d
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.