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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 38720.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38720.bm1 | 38720bu4 | \([0, 0, 0, -51788, -4536048]\) | \(132304644/5\) | \(580505108480\) | \([2]\) | \(81920\) | \(1.3433\) | |
| 38720.bm2 | 38720bu2 | \([0, 0, 0, -3388, -63888]\) | \(148176/25\) | \(725631385600\) | \([2, 2]\) | \(40960\) | \(0.99673\) | |
| 38720.bm3 | 38720bu1 | \([0, 0, 0, -968, 10648]\) | \(55296/5\) | \(9070392320\) | \([2]\) | \(20480\) | \(0.65016\) | \(\Gamma_0(N)\)-optimal |
| 38720.bm4 | 38720bu3 | \([0, 0, 0, 6292, -362032]\) | \(237276/625\) | \(-72563138560000\) | \([2]\) | \(81920\) | \(1.3433\) |
Rank
sage: E.rank()
The elliptic curves in class 38720.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 38720.bm do not have complex multiplication.Modular form 38720.2.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.
