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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 650.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
650.d1 | 650a3 | \([1, -1, 0, -34667, -2475759]\) | \(294889639316481/260\) | \(4062500\) | \([2]\) | \(768\) | \(1.0005\) | |
650.d2 | 650a2 | \([1, -1, 0, -2167, -38259]\) | \(72043225281/67600\) | \(1056250000\) | \([2, 2]\) | \(384\) | \(0.65393\) | |
650.d3 | 650a4 | \([1, -1, 0, -1667, -56759]\) | \(-32798729601/71402500\) | \(-1115664062500\) | \([2]\) | \(768\) | \(1.0005\) | |
650.d4 | 650a1 | \([1, -1, 0, -167, -259]\) | \(33076161/16640\) | \(260000000\) | \([2]\) | \(192\) | \(0.30736\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 650.d have rank \(1\).
Complex multiplication
The elliptic curves in class 650.d do not have complex multiplication.Modular form 650.2.a.d
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.