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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 33600.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.k1 | 33600k4 | \([0, -1, 0, -68033, -6804063]\) | \(68017239368/39375\) | \(20160000000000\) | \([2]\) | \(98304\) | \(1.4985\) | |
| 33600.k2 | 33600k3 | \([0, -1, 0, -40033, 3051937]\) | \(13858588808/229635\) | \(117573120000000\) | \([2]\) | \(98304\) | \(1.4985\) | |
| 33600.k3 | 33600k2 | \([0, -1, 0, -5033, -63063]\) | \(220348864/99225\) | \(6350400000000\) | \([2, 2]\) | \(49152\) | \(1.1519\) | |
| 33600.k4 | 33600k1 | \([0, -1, 0, 1092, -7938]\) | \(143877824/108045\) | \(-108045000000\) | \([2]\) | \(24576\) | \(0.80534\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.k have rank \(1\).
Complex multiplication
The elliptic curves in class 33600.k do not have complex multiplication.Modular form 33600.2.a.k
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.
