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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1728e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1728.c2 | 1728e1 | \([0, 0, 0, -204, 1136]\) | \(-132651/2\) | \(-14155776\) | \([]\) | \(384\) | \(0.17521\) | \(\Gamma_0(N)\)-optimal |
| 1728.c3 | 1728e2 | \([0, 0, 0, 756, 5616]\) | \(9261/8\) | \(-41278242816\) | \([]\) | \(1152\) | \(0.72452\) | |
| 1728.c1 | 1728e3 | \([0, 0, 0, -7884, -357264]\) | \(-1167051/512\) | \(-23776267862016\) | \([]\) | \(3456\) | \(1.2738\) |
Rank
sage: E.rank()
The elliptic curves in class 1728e have rank \(1\).
Complex multiplication
The elliptic curves in class 1728e do not have complex multiplication.Modular form 1728.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
