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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 330d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330.c3 | 330d1 | \([1, 1, 1, -40266, 2921559]\) | \(7220044159551112609/448454983680000\) | \(448454983680000\) | \([4]\) | \(2240\) | \(1.5629\) | \(\Gamma_0(N)\)-optimal |
| 330.c2 | 330d2 | \([1, 1, 1, -122186, -12872617]\) | \(201738262891771037089/45727545600000000\) | \(45727545600000000\) | \([2, 2]\) | \(4480\) | \(1.9095\) | |
| 330.c1 | 330d3 | \([1, 1, 1, -1832906, -955821481]\) | \(680995599504466943307169/52207031250000000\) | \(52207031250000000\) | \([2]\) | \(8960\) | \(2.2561\) | |
| 330.c4 | 330d4 | \([1, 1, 1, 277814, -79112617]\) | \(2371297246710590562911/4084000833203280000\) | \(-4084000833203280000\) | \([2]\) | \(8960\) | \(2.2561\) |
Rank
sage: E.rank()
The elliptic curves in class 330d have rank \(0\).
Complex multiplication
The elliptic curves in class 330d do not have complex multiplication.Modular form 330.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
