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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 54450.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54450.d1 | 54450cq4 | \([1, -1, 0, -9583767, -11417246109]\) | \(4824238966273/66\) | \(1331826343031250\) | \([2]\) | \(1966080\) | \(2.4580\) | |
| 54450.d2 | 54450cq2 | \([1, -1, 0, -599517, -177949359]\) | \(1180932193/4356\) | \(87900538640062500\) | \([2, 2]\) | \(983040\) | \(2.1114\) | |
| 54450.d3 | 54450cq3 | \([1, -1, 0, -327267, -340482609]\) | \(-192100033/2371842\) | \(-47861843289514031250\) | \([2]\) | \(1966080\) | \(2.4580\) | |
| 54450.d4 | 54450cq1 | \([1, -1, 0, -55017, 102141]\) | \(912673/528\) | \(10654610744250000\) | \([2]\) | \(491520\) | \(1.7649\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54450.d have rank \(1\).
Complex multiplication
The elliptic curves in class 54450.d do not have complex multiplication.Modular form 54450.2.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.
