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SageMath
E = EllipticCurve("np1")
E.isogeny_class()
Elliptic curves in class 366912.np
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 366912.np1 | 366912np4 | \([0, 0, 0, -6605004, -6533678032]\) | \(11339065490696/351\) | \(986444872777728\) | \([2]\) | \(9437184\) | \(2.3815\) | |
| 366912.np2 | 366912np2 | \([0, 0, 0, -413364, -101802400]\) | \(22235451328/123201\) | \(43280268793122816\) | \([2, 2]\) | \(4718592\) | \(2.0350\) | |
| 366912.np3 | 366912np3 | \([0, 0, 0, -184044, -214169200]\) | \(-245314376/6908733\) | \(-19416194430884020224\) | \([2]\) | \(9437184\) | \(2.3815\) | |
| 366912.np4 | 366912np1 | \([0, 0, 0, -40719, 451388]\) | \(1360251712/771147\) | \(4232850362290368\) | \([2]\) | \(2359296\) | \(1.6884\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 366912.np have rank \(0\).
Complex multiplication
The elliptic curves in class 366912.np do not have complex multiplication.Modular form 366912.2.a.np
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.
