Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 68970.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68970.cf1 | 68970cq4 | \([1, 0, 0, -149246666, 701771808420]\) | \(207530301091125281552569/805586668007040\) | \(1427145923161219789440\) | \([2]\) | \(12902400\) | \(3.2724\) | |
| 68970.cf2 | 68970cq3 | \([1, 0, 0, -28285386, -44708872284]\) | \(1412712966892699019449/330160465517040000\) | \(584899404451832899440000\) | \([2]\) | \(12902400\) | \(3.2724\) | |
| 68970.cf3 | 68970cq2 | \([1, 0, 0, -9467466, 10619576100]\) | \(52974743974734147769/3152005008998400\) | \(5583969145746214502400\) | \([2, 2]\) | \(6451200\) | \(2.9258\) | |
| 68970.cf4 | 68970cq1 | \([1, 0, 0, 444854, 685448996]\) | \(5495662324535111/117739817533440\) | \(-208583268889358499840\) | \([2]\) | \(3225600\) | \(2.5792\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68970.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 68970.cf do not have complex multiplication.Modular form 68970.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
