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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 29760.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29760.ck1 | 29760cx4 | \([0, 1, 0, -96865, 11071775]\) | \(383432500775449/18701300250\) | \(4902433652736000\) | \([4]\) | \(221184\) | \(1.7703\) | |
| 29760.ck2 | 29760cx2 | \([0, 1, 0, -16865, -624225]\) | \(2023804595449/540562500\) | \(141705216000000\) | \([2, 2]\) | \(110592\) | \(1.4237\) | |
| 29760.ck3 | 29760cx1 | \([0, 1, 0, -15585, -754017]\) | \(1597099875769/186000\) | \(48758784000\) | \([2]\) | \(55296\) | \(1.0772\) | \(\Gamma_0(N)\)-optimal |
| 29760.ck4 | 29760cx3 | \([0, 1, 0, 42655, -3993057]\) | \(32740359775271/45410156250\) | \(-11904000000000000\) | \([2]\) | \(221184\) | \(1.7703\) |
Rank
sage: E.rank()
The elliptic curves in class 29760.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 29760.ck do not have complex multiplication.Modular form 29760.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
