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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 42042n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.c2 | 42042n1 | \([1, 1, 0, 514, 63540]\) | \(127263527/14798784\) | \(-1741062138816\) | \([2]\) | \(73728\) | \(1.0282\) | \(\Gamma_0(N)\)-optimal |
42042.c1 | 42042n2 | \([1, 1, 0, -21046, 1128604]\) | \(8763476225113/321369048\) | \(37808747128152\) | \([2]\) | \(147456\) | \(1.3747\) |
Rank
sage: E.rank()
The elliptic curves in class 42042n have rank \(1\).
Complex multiplication
The elliptic curves in class 42042n do not have complex multiplication.Modular form 42042.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.