Show commands:
SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 87120.fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
87120.fd1 | 87120fp2 | \([0, 0, 0, -104392992, -410540433424]\) | \(-196566176333824/421875\) | \(-270030454702272000000\) | \([]\) | \(8211456\) | \(3.1679\) | |
87120.fd2 | 87120fp1 | \([0, 0, 0, -894432, -913832656]\) | \(-123633664/492075\) | \(-314963522364730060800\) | \([]\) | \(2737152\) | \(2.6186\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 87120.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 87120.fd do not have complex multiplication.Modular form 87120.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.