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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4017.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4017.a1 | 4017a2 | \([1, 1, 1, -542, 4628]\) | \(17610498546913/12689703\) | \(12689703\) | \([2]\) | \(1248\) | \(0.29805\) | |
4017.a2 | 4017a1 | \([1, 1, 1, -27, 96]\) | \(-2181825073/3723759\) | \(-3723759\) | \([2]\) | \(624\) | \(-0.048528\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4017.a have rank \(1\).
Complex multiplication
The elliptic curves in class 4017.a do not have complex multiplication.Modular form 4017.2.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.