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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 73920fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.cp3 | 73920fd1 | \([0, -1, 0, -13025, -567423]\) | \(932288503609/779625\) | \(204374016000\) | \([2]\) | \(147456\) | \(1.0982\) | \(\Gamma_0(N)\)-optimal |
| 73920.cp2 | 73920fd2 | \([0, -1, 0, -15905, -294975]\) | \(1697509118089/833765625\) | \(218566656000000\) | \([2, 2]\) | \(294912\) | \(1.4447\) | |
| 73920.cp4 | 73920fd3 | \([0, -1, 0, 58015, -2320383]\) | \(82375335041831/56396484375\) | \(-14784000000000000\) | \([2]\) | \(589824\) | \(1.7913\) | |
| 73920.cp1 | 73920fd4 | \([0, -1, 0, -135905, 19121025]\) | \(1058993490188089/13182390375\) | \(3455684542464000\) | \([2]\) | \(589824\) | \(1.7913\) |
Rank
sage: E.rank()
The elliptic curves in class 73920fd have rank \(1\).
Complex multiplication
The elliptic curves in class 73920fd do not have complex multiplication.Modular form 73920.2.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 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.
