Show commands:
SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 221067.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221067.bb1 | 221067z4 | \([1, -1, 0, -1360728, -610209275]\) | \(215751695207833/163381911\) | \(211002504770508759\) | \([2]\) | \(3194880\) | \(2.2569\) | |
| 221067.bb2 | 221067z2 | \([1, -1, 0, -102933, -5209880]\) | \(93391282153/44876601\) | \(57956692749093369\) | \([2, 2]\) | \(1597440\) | \(1.9103\) | |
| 221067.bb3 | 221067z1 | \([1, -1, 0, -53928, 4777339]\) | \(13430356633/180873\) | \(233591685956937\) | \([2]\) | \(798720\) | \(1.5637\) | \(\Gamma_0(N)\)-optimal |
| 221067.bb4 | 221067z3 | \([1, -1, 0, 370782, -39980561]\) | \(4365111505607/3058314567\) | \(-3949715302406604423\) | \([2]\) | \(3194880\) | \(2.2569\) |
Rank
sage: E.rank()
The elliptic curves in class 221067.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 221067.bb do not have complex multiplication.Modular form 221067.2.a.bb
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.
