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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 66.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66.b1 | 66b3 | \([1, 1, 1, -352, -2689]\) | \(4824238966273/66\) | \(66\) | \([2]\) | \(16\) | \(-0.094954\) | |
66.b2 | 66b2 | \([1, 1, 1, -22, -49]\) | \(1180932193/4356\) | \(4356\) | \([2, 2]\) | \(8\) | \(-0.44153\) | |
66.b3 | 66b4 | \([1, 1, 1, -12, -81]\) | \(-192100033/2371842\) | \(-2371842\) | \([2]\) | \(16\) | \(-0.094954\) | |
66.b4 | 66b1 | \([1, 1, 1, -2, -1]\) | \(912673/528\) | \(528\) | \([4]\) | \(4\) | \(-0.78810\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66.b have rank \(0\).
Complex multiplication
The elliptic curves in class 66.b do not have complex multiplication.Modular form 66.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.