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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 156702.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
156702.b1 | 156702cl4 | \([1, 1, 0, -72741, -7271025]\) | \(361811696411593/16869440406\) | \(1984672794325494\) | \([2]\) | \(1499136\) | \(1.6965\) | |
156702.b2 | 156702cl2 | \([1, 1, 0, -12471, 383265]\) | \(1823449422313/501132996\) | \(58957795846404\) | \([2, 2]\) | \(749568\) | \(1.3499\) | |
156702.b3 | 156702cl1 | \([1, 1, 0, -11491, 469309]\) | \(1426487591593/179088\) | \(21069524112\) | \([2]\) | \(374784\) | \(1.0033\) | \(\Gamma_0(N)\)-optimal |
156702.b4 | 156702cl3 | \([1, 1, 0, 32119, 2550339]\) | \(31145864569847/41657368662\) | \(-4900947765715638\) | \([2]\) | \(1499136\) | \(1.6965\) |
Rank
sage: E.rank()
The elliptic curves in class 156702.b have rank \(1\).
Complex multiplication
The elliptic curves in class 156702.b do not have complex multiplication.Modular form 156702.2.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.