Show commands:
SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 106722.bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.bu1 | 106722cu3 | \([1, -1, 0, -4296672, 3415983808]\) | \(57736239625/255552\) | \(38828549179771509312\) | \([2]\) | \(4147200\) | \(2.6111\) | |
| 106722.bu2 | 106722cu4 | \([1, -1, 0, -2162232, 6809316520]\) | \(-7357983625/127552392\) | \(-19380299609353454585352\) | \([2]\) | \(8294400\) | \(2.9577\) | |
| 106722.bu3 | 106722cu1 | \([1, -1, 0, -294597, -57977375]\) | \(18609625/1188\) | \(180504619120838628\) | \([2]\) | \(1382400\) | \(2.0618\) | \(\Gamma_0(N)\)-optimal |
| 106722.bu4 | 106722cu2 | \([1, -1, 0, 239013, -245061041]\) | \(9938375/176418\) | \(-26804935939444536258\) | \([2]\) | \(2764800\) | \(2.4084\) |
Rank
sage: E.rank()
The elliptic curves in class 106722.bu have rank \(2\).
Complex multiplication
The elliptic curves in class 106722.bu do not have complex multiplication.Modular form 106722.2.a.bu
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.
