Show commands:
SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 50715bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 50715.bq3 | 50715bq1 | \([1, -1, 0, -123489, 16685248]\) | \(2428257525121/8150625\) | \(699047489975625\) | \([2]\) | \(196608\) | \(1.7127\) | \(\Gamma_0(N)\)-optimal |
| 50715.bq2 | 50715bq2 | \([1, -1, 0, -178614, 357223]\) | \(7347774183121/4251692025\) | \(364651132670885025\) | \([2, 2]\) | \(393216\) | \(2.0593\) | |
| 50715.bq4 | 50715bq3 | \([1, -1, 0, 714411, 2321878]\) | \(470166844956479/272118787605\) | \(-23338572864103730205\) | \([2]\) | \(786432\) | \(2.4059\) | |
| 50715.bq1 | 50715bq4 | \([1, -1, 0, -1953639, -1047262532]\) | \(9614816895690721/34652610405\) | \(2972019976961089005\) | \([2]\) | \(786432\) | \(2.4059\) |
Rank
sage: E.rank()
The elliptic curves in class 50715bq have rank \(1\).
Complex multiplication
The elliptic curves in class 50715bq do not have complex multiplication.Modular form 50715.2.a.bq
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 Cremona 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 Cremona labels.
