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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 471510.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.n1 | 471510n4 | \([1, -1, 0, -10081980, -12318855080]\) | \(32208729120020809/658986840\) | \(2318805831831105240\) | \([2]\) | \(17694720\) | \(2.6435\) | |
471510.n2 | 471510n2 | \([1, -1, 0, -651780, -178415600]\) | \(8702409880009/1120910400\) | \(3944196476640014400\) | \([2, 2]\) | \(8847360\) | \(2.2969\) | |
471510.n3 | 471510n1 | \([1, -1, 0, -165060, 22989136]\) | \(141339344329/17141760\) | \(60317461052559360\) | \([2]\) | \(4423680\) | \(1.9504\) | \(\Gamma_0(N)\)-optimal* |
471510.n4 | 471510n3 | \([1, -1, 0, 990900, -933719864]\) | \(30579142915511/124675335000\) | \(-438700557181834935000\) | \([2]\) | \(17694720\) | \(2.6435\) |
Rank
sage: E.rank()
The elliptic curves in class 471510.n have rank \(1\).
Complex multiplication
The elliptic curves in class 471510.n do not have complex multiplication.Modular form 471510.2.a.n
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.