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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 361998.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 361998.p1 | 361998p3 | \([1, -1, 0, -4717413, 3311612019]\) | \(3299497626614617/563987509722\) | \(1984527531116214344442\) | \([2]\) | \(20643840\) | \(2.8071\) | |
| 361998.p2 | 361998p2 | \([1, -1, 0, -1356003, -558715455]\) | \(78364289651257/6978597444\) | \(24555896216605546884\) | \([2, 2]\) | \(10321920\) | \(2.4606\) | |
| 361998.p3 | 361998p1 | \([1, -1, 0, -1325583, -587097315]\) | \(73207745356537/668304\) | \(2351590530451344\) | \([2]\) | \(5160960\) | \(2.1140\) | \(\Gamma_0(N)\)-optimal |
| 361998.p4 | 361998p4 | \([1, -1, 0, 1518687, -2612968929]\) | \(110088190986983/901697560218\) | \(-3172842664326009299898\) | \([2]\) | \(20643840\) | \(2.8071\) |
Rank
sage: E.rank()
The elliptic curves in class 361998.p have rank \(1\).
Complex multiplication
The elliptic curves in class 361998.p do not have complex multiplication.Modular form 361998.2.a.p
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.
