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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 477425.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
477425.a1 | 477425a1 | \([1, 0, 0, -8538, -269933]\) | \(912673/113\) | \(8522334640625\) | \([2]\) | \(884736\) | \(1.2110\) | \(\Gamma_0(N)\)-optimal |
477425.a2 | 477425a2 | \([1, 0, 0, 12587, -1389558]\) | \(2924207/12769\) | \(-963023814390625\) | \([2]\) | \(1769472\) | \(1.5576\) |
Rank
sage: E.rank()
The elliptic curves in class 477425.a have rank \(1\).
Complex multiplication
The elliptic curves in class 477425.a do not have complex multiplication.Modular form 477425.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.