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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 258570.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.bm1 | 258570bm3 | \([1, -1, 0, -2127150, 1152231250]\) | \(302503589987689/12214946250\) | \(42981265908137846250\) | \([2]\) | \(11010048\) | \(2.5334\) | |
| 258570.bm2 | 258570bm2 | \([1, -1, 0, -347580, -54673124]\) | \(1319778683209/395612100\) | \(1392057608651108100\) | \([2, 2]\) | \(5505024\) | \(2.1868\) | |
| 258570.bm3 | 258570bm1 | \([1, -1, 0, -317160, -68660240]\) | \(1002702430729/159120\) | \(559902507250320\) | \([2]\) | \(2752512\) | \(1.8402\) | \(\Gamma_0(N)\)-optimal |
| 258570.bm4 | 258570bm4 | \([1, -1, 0, 945270, -366767114]\) | \(26546265663191/31856082570\) | \(-112093391793088345770\) | \([2]\) | \(11010048\) | \(2.5334\) |
Rank
sage: E.rank()
The elliptic curves in class 258570.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 258570.bm do not have complex multiplication.Modular form 258570.2.a.bm
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.
