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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 39600.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39600.bm1 | 39600dz3 | \([0, 0, 0, -36234075, -83950613750]\) | \(112763292123580561/1932612\) | \(90167945472000000\) | \([2]\) | \(1536000\) | \(2.7950\) | |
| 39600.bm2 | 39600dz4 | \([0, 0, 0, -36198075, -84125753750]\) | \(-112427521449300721/466873642818\) | \(-21782456679316608000000\) | \([2]\) | \(3072000\) | \(3.1416\) | |
| 39600.bm3 | 39600dz1 | \([0, 0, 0, -162075, 18306250]\) | \(10091699281/2737152\) | \(127704563712000000\) | \([2]\) | \(307200\) | \(1.9903\) | \(\Gamma_0(N)\)-optimal |
| 39600.bm4 | 39600dz2 | \([0, 0, 0, 413925, 119106250]\) | \(168105213359/228637728\) | \(-10667321837568000000\) | \([2]\) | \(614400\) | \(2.3369\) |
Rank
sage: E.rank()
The elliptic curves in class 39600.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 39600.bm do not have complex multiplication.Modular form 39600.2.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.
