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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 67270.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67270.bd1 | 67270y4 | \([1, -1, 1, -257248, 50281697]\) | \(2121328796049/120050\) | \(106544816904050\) | \([2]\) | \(460800\) | \(1.7560\) | |
| 67270.bd2 | 67270y3 | \([1, -1, 1, -84268, -8777519]\) | \(74565301329/5468750\) | \(4853535755468750\) | \([2]\) | \(460800\) | \(1.7560\) | |
| 67270.bd3 | 67270y2 | \([1, -1, 1, -16998, 694097]\) | \(611960049/122500\) | \(108719200922500\) | \([2, 2]\) | \(230400\) | \(1.4094\) | |
| 67270.bd4 | 67270y1 | \([1, -1, 1, 2222, 63681]\) | \(1367631/2800\) | \(-2485010306800\) | \([2]\) | \(115200\) | \(1.0628\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67270.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 67270.bd do not have complex multiplication.Modular form 67270.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
