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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 47040fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.dg1 | 47040fd1 | \([0, -1, 0, -1045, 5125]\) | \(1048576/525\) | \(63248102400\) | \([2]\) | \(36864\) | \(0.76532\) | \(\Gamma_0(N)\)-optimal |
| 47040.dg2 | 47040fd2 | \([0, -1, 0, 3855, 35505]\) | \(3286064/2205\) | \(-4250272481280\) | \([2]\) | \(73728\) | \(1.1119\) |
Rank
sage: E.rank()
The elliptic curves in class 47040fd have rank \(0\).
Complex multiplication
The elliptic curves in class 47040fd do not have complex multiplication.Modular form 47040.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
