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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 28224ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.fu2 | 28224ex1 | \([0, 0, 0, 13524, -742448]\) | \(596183/864\) | \(-396436244004864\) | \([]\) | \(92160\) | \(1.4866\) | \(\Gamma_0(N)\)-optimal |
| 28224.fu1 | 28224ex2 | \([0, 0, 0, -409836, -101502128]\) | \(-16591834777/98304\) | \(-45105634873442304\) | \([]\) | \(276480\) | \(2.0359\) |
Rank
sage: E.rank()
The elliptic curves in class 28224ex have rank \(1\).
Complex multiplication
The elliptic curves in class 28224ex do not have complex multiplication.Modular form 28224.2.a.ex
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
