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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 84150.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.bm1 | 84150ba4 | \([1, -1, 0, -2247417, -1296230009]\) | \(110211585818155849/993794670\) | \(11319942412968750\) | \([2]\) | \(1769472\) | \(2.2456\) | |
| 84150.bm2 | 84150ba2 | \([1, -1, 0, -143667, -19253759]\) | \(28790481449449/2549240100\) | \(29037438014062500\) | \([2, 2]\) | \(884736\) | \(1.8991\) | |
| 84150.bm3 | 84150ba1 | \([1, -1, 0, -31167, 1783741]\) | \(293946977449/50490000\) | \(575112656250000\) | \([2]\) | \(442368\) | \(1.5525\) | \(\Gamma_0(N)\)-optimal |
| 84150.bm4 | 84150ba3 | \([1, -1, 0, 160083, -90027509]\) | \(39829997144951/330164359470\) | \(-3760778407087968750\) | \([2]\) | \(1769472\) | \(2.2456\) |
Rank
sage: E.rank()
The elliptic curves in class 84150.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 84150.bm do not have complex multiplication.Modular form 84150.2.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.
