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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 89232cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89232.bk4 | 89232cd1 | \([0, 1, 0, -5464, -4588]\) | \(912673/528\) | \(10438881902592\) | \([2]\) | \(184320\) | \(1.1875\) | \(\Gamma_0(N)\)-optimal |
| 89232.bk2 | 89232cd2 | \([0, 1, 0, -59544, 5554836]\) | \(1180932193/4356\) | \(86120775696384\) | \([2, 2]\) | \(368640\) | \(1.5341\) | |
| 89232.bk3 | 89232cd3 | \([0, 1, 0, -32504, 10649172]\) | \(-192100033/2371842\) | \(-46892762366681088\) | \([2]\) | \(737280\) | \(1.8807\) | |
| 89232.bk1 | 89232cd4 | \([0, 1, 0, -951864, 357128916]\) | \(4824238966273/66\) | \(1304860237824\) | \([2]\) | \(737280\) | \(1.8807\) |
Rank
sage: E.rank()
The elliptic curves in class 89232cd have rank \(1\).
Complex multiplication
The elliptic curves in class 89232cd do not have complex multiplication.Modular form 89232.2.a.cd
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.
