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SageMath
sage: E = EllipticCurve("pb1")
sage: E.isogeny_class()
Elliptic curves in class 369600.pb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 369600.pb1 | 369600pb6 | [0, 1, 0, -4878720033, -131163302591937] | [2] | 94371840 | |
| 369600.pb2 | 369600pb4 | [0, 1, 0, -304920033, -2049502391937] | [2, 2] | 47185920 | |
| 369600.pb3 | 369600pb5 | [0, 1, 0, -303408033, -2070832175937] | [2] | 94371840 | |
| 369600.pb4 | 369600pb3 | [0, 1, 0, -40712033, 52712856063] | [2] | 47185920 | |
| 369600.pb5 | 369600pb2 | [0, 1, 0, -19152033, -31694543937] | [2, 2] | 23592960 | |
| 369600.pb6 | 369600pb1 | [0, 1, 0, 55967, -1480359937] | [2] | 11796480 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.pb have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.pb do not have complex multiplication.Modular form 369600.2.a.pb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
