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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 52800.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52800.bm1 | 52800fb3 | \([0, -1, 0, -16104033, 24879623937]\) | \(112763292123580561/1932612\) | \(7915978752000000\) | \([2]\) | \(1536000\) | \(2.5923\) | |
| 52800.bm2 | 52800fb4 | \([0, -1, 0, -16088033, 24931511937]\) | \(-112427521449300721/466873642818\) | \(-1912314440982528000000\) | \([2]\) | \(3072000\) | \(2.9389\) | |
| 52800.bm3 | 52800fb1 | \([0, -1, 0, -72033, -5400063]\) | \(10091699281/2737152\) | \(11211374592000000\) | \([2]\) | \(307200\) | \(1.7876\) | \(\Gamma_0(N)\)-optimal |
| 52800.bm4 | 52800fb2 | \([0, -1, 0, 183967, -35352063]\) | \(168105213359/228637728\) | \(-936500133888000000\) | \([2]\) | \(614400\) | \(2.1341\) |
Rank
sage: E.rank()
The elliptic curves in class 52800.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 52800.bm do not have complex multiplication.Modular form 52800.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
