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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 198d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198.b4 | 198d1 | \([1, -1, 0, -87, 333]\) | \(2714704875/21296\) | \(574992\) | \([6]\) | \(32\) | \(-0.066515\) | \(\Gamma_0(N)\)-optimal |
| 198.b3 | 198d2 | \([1, -1, 0, -147, -135]\) | \(13060888875/7086244\) | \(191328588\) | \([6]\) | \(64\) | \(0.28006\) | |
| 198.b2 | 198d3 | \([1, -1, 0, -582, -5068]\) | \(1108717875/45056\) | \(886837248\) | \([2]\) | \(96\) | \(0.48279\) | |
| 198.b1 | 198d4 | \([1, -1, 0, -9222, -338572]\) | \(4406910829875/7744\) | \(152425152\) | \([2]\) | \(192\) | \(0.82936\) |
Rank
sage: E.rank()
The elliptic curves in class 198d have rank \(0\).
Complex multiplication
The elliptic curves in class 198d do not have complex multiplication.Modular form 198.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
