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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 58800.fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.fd1 | 58800ge4 | \([0, -1, 0, -2239708, -1289387588]\) | \(2640279346000/3087\) | \(1452729852000000\) | \([2]\) | \(995328\) | \(2.1925\) | |
| 58800.fd2 | 58800ge3 | \([0, -1, 0, -138833, -20459088]\) | \(-10061824000/352947\) | \(-10380965400750000\) | \([2]\) | \(497664\) | \(1.8459\) | |
| 58800.fd3 | 58800ge2 | \([0, -1, 0, -34708, -785588]\) | \(9826000/5103\) | \(2401451388000000\) | \([2]\) | \(331776\) | \(1.6432\) | |
| 58800.fd4 | 58800ge1 | \([0, -1, 0, 8167, -99588]\) | \(2048000/1323\) | \(-38912406750000\) | \([2]\) | \(165888\) | \(1.2966\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58800.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.fd do not have complex multiplication.Modular form 58800.2.a.fd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
