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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1984.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1984.f1 | 1984b4 | \([0, 0, 0, -21164, 1185072]\) | \(3999236143617/62\) | \(16252928\) | \([4]\) | \(1536\) | \(0.93164\) | |
| 1984.f2 | 1984b3 | \([0, 0, 0, -1964, -1232]\) | \(3196010817/1847042\) | \(484190978048\) | \([2]\) | \(1536\) | \(0.93164\) | |
| 1984.f3 | 1984b2 | \([0, 0, 0, -1324, 18480]\) | \(979146657/3844\) | \(1007681536\) | \([2, 2]\) | \(768\) | \(0.58507\) | |
| 1984.f4 | 1984b1 | \([0, 0, 0, -44, 560]\) | \(-35937/496\) | \(-130023424\) | \([2]\) | \(384\) | \(0.23850\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1984.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1984.f do not have complex multiplication.Modular form 1984.2.a.f
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.
