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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 178464.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 178464.ck1 | 178464bw2 | \([0, 1, 0, -89457, 10268415]\) | \(4004529472/99\) | \(1957290356736\) | \([2]\) | \(491520\) | \(1.4677\) | |
| 178464.ck2 | 178464bw4 | \([0, 1, 0, -24392, -1326168]\) | \(649461896/72171\) | \(178358083757568\) | \([2]\) | \(491520\) | \(1.4677\) | |
| 178464.ck3 | 178464bw1 | \([0, 1, 0, -5802, 146160]\) | \(69934528/9801\) | \(3027683520576\) | \([2, 2]\) | \(245760\) | \(1.1211\) | \(\Gamma_0(N)\)-optimal |
| 178464.ck4 | 178464bw3 | \([0, 1, 0, 9408, 797148]\) | \(37259704/131769\) | \(-325644183101952\) | \([2]\) | \(491520\) | \(1.4677\) |
Rank
sage: E.rank()
The elliptic curves in class 178464.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 178464.ck do not have complex multiplication.Modular form 178464.2.a.ck
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.
