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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 33600.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.cp1 | 33600bc4 | \([0, -1, 0, -8400033, -9367848063]\) | \(128025588102048008/7875\) | \(4032000000000\) | \([2]\) | \(589824\) | \(2.3273\) | |
| 33600.cp2 | 33600bc3 | \([0, -1, 0, -588033, -108864063]\) | \(43919722445768/15380859375\) | \(7875000000000000000\) | \([2]\) | \(589824\) | \(2.3273\) | |
| 33600.cp3 | 33600bc2 | \([0, -1, 0, -525033, -146223063]\) | \(250094631024064/62015625\) | \(3969000000000000\) | \([2, 2]\) | \(294912\) | \(1.9807\) | |
| 33600.cp4 | 33600bc1 | \([0, -1, 0, -28908, -2842938]\) | \(-2671731885376/1969120125\) | \(-1969120125000000\) | \([2]\) | \(147456\) | \(1.6341\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.cp do not have complex multiplication.Modular form 33600.2.a.cp
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.
