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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 39.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39.a1 | 39a2 | \([1, 1, 0, -69, -252]\) | \(37159393753/1053\) | \(1053\) | \([2]\) | \(4\) | \(-0.31837\) | |
39.a2 | 39a3 | \([1, 1, 0, -19, 22]\) | \(822656953/85683\) | \(85683\) | \([4]\) | \(4\) | \(-0.31837\) | |
39.a3 | 39a1 | \([1, 1, 0, -4, -5]\) | \(10218313/1521\) | \(1521\) | \([2, 2]\) | \(2\) | \(-0.66494\) | \(\Gamma_0(N)\)-optimal |
39.a4 | 39a4 | \([1, 1, 0, 1, 0]\) | \(12167/39\) | \(-39\) | \([2]\) | \(4\) | \(-1.0115\) |
Rank
sage: E.rank()
The elliptic curves in class 39.a have rank \(0\).
Complex multiplication
The elliptic curves in class 39.a do not have complex multiplication.Modular form 39.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.