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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 16650.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16650.bk1 | 16650ba3 | \([1, -1, 0, -80621574417, 8811016602171741]\) | \(5087799435928552778197163696329/125914832087040\) | \(1434248634241440000000\) | \([2]\) | \(37847040\) | \(4.5052\) | |
| 16650.bk2 | 16650ba2 | \([1, -1, 0, -5038854417, 137672734011741]\) | \(1242142983306846366056931529/6179359141291622400\) | \(70386762718774886400000000\) | \([2, 2]\) | \(18923520\) | \(4.1586\) | |
| 16650.bk3 | 16650ba4 | \([1, -1, 0, -4953606417, 142555654203741]\) | \(-1180159344892952613848670409/87759036144023189760000\) | \(-999630271078014145860000000000\) | \([2]\) | \(37847040\) | \(4.5052\) | |
| 16650.bk4 | 16650ba1 | \([1, -1, 0, -320262417, 2074555707741]\) | \(318929057401476905525449/21353131537921474560\) | \(243225513924136796160000000\) | \([2]\) | \(9461760\) | \(3.8121\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 16650.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 16650.bk do not have complex multiplication.Modular form 16650.2.a.bk
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.
