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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 77616.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.bf1 | 77616fk6 | \([0, 0, 0, -31886211, 69303039554]\) | \(10206027697760497/5557167\) | \(1952221828461391872\) | \([2]\) | \(3932160\) | \(2.8382\) | |
| 77616.bf2 | 77616fk4 | \([0, 0, 0, -2004051, 1070115410]\) | \(2533811507137/58110129\) | \(20413973934651838464\) | \([2, 2]\) | \(1966080\) | \(2.4916\) | |
| 77616.bf3 | 77616fk2 | \([0, 0, 0, -275331, -31079230]\) | \(6570725617/2614689\) | \(918535098988007424\) | \([2, 2]\) | \(983040\) | \(2.1451\) | |
| 77616.bf4 | 77616fk1 | \([0, 0, 0, -240051, -45254734]\) | \(4354703137/1617\) | \(568048917123072\) | \([2]\) | \(491520\) | \(1.7985\) | \(\Gamma_0(N)\)-optimal |
| 77616.bf5 | 77616fk5 | \([0, 0, 0, 218589, 3313648226]\) | \(3288008303/13504609503\) | \(-4744142736146628046848\) | \([2]\) | \(3932160\) | \(2.8382\) | |
| 77616.bf6 | 77616fk3 | \([0, 0, 0, 888909, -225041614]\) | \(221115865823/190238433\) | \(-66830387050612297728\) | \([2]\) | \(1966080\) | \(2.4916\) |
Rank
sage: E.rank()
The elliptic curves in class 77616.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 77616.bf do not have complex multiplication.Modular form 77616.2.a.bf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
