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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 19890.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19890.f1 | 19890f2 | \([1, -1, 0, -21960, 1226016]\) | \(1606587247762561/46962500000\) | \(34235662500000\) | \([2]\) | \(61440\) | \(1.3745\) | |
| 19890.f2 | 19890f1 | \([1, -1, 0, -3240, -43200]\) | \(5160676199041/1838720000\) | \(1340426880000\) | \([2]\) | \(30720\) | \(1.0280\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19890.f have rank \(1\).
Complex multiplication
The elliptic curves in class 19890.f do not have complex multiplication.Modular form 19890.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
