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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 166635.bx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 166635.bx1 | 166635bh4 | \([1, -1, 0, -21091329, -37160866260]\) | \(9614816895690721/34652610405\) | \(3739646060869147457805\) | \([2]\) | \(8650752\) | \(3.0006\) | |
| 166635.bx2 | 166635bh2 | \([1, -1, 0, -1928304, 11569635]\) | \(7347774183121/4251692025\) | \(458834793324137129025\) | \([2, 2]\) | \(4325376\) | \(2.6541\) | |
| 166635.bx3 | 166635bh1 | \([1, -1, 0, -1333179, 591102360]\) | \(2428257525121/8150625\) | \(879600477962075625\) | \([2]\) | \(2162688\) | \(2.3075\) | \(\Gamma_0(N)\)-optimal |
| 166635.bx4 | 166635bh3 | \([1, -1, 0, 7712721, 86769630]\) | \(470166844956479/272118787605\) | \(-29366559698160391411005\) | \([2]\) | \(8650752\) | \(3.0006\) |
Rank
sage: E.rank()
The elliptic curves in class 166635.bx have rank \(1\).
Complex multiplication
The elliptic curves in class 166635.bx do not have complex multiplication.Modular form 166635.2.a.bx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
